title: Median

text: Note that this definition does not require "X" to have an absolutely continuous distribution (which has a probability density function "f"), nor does it require a discrete one. In the former case, the inequalities can be upgraded to equality: a median satisfies


title: Median

text: Any probability distribution on R has at least one median, but in pathological cases there may be more than one median: if "F" is constant 1/2 on an interval (so that "f"=0 there), then any value of that interval is a median.


title: Median

text: The medians of certain types of distributions can be easily calculated from their parameters; furthermore, they exist even for some distributions lacking a well-defined mean, such as the Cauchy distribution:


title: Median

text: Provided that the probability distribution of "X" is such that the above expectation exists, then "m" is a median of "X" if and only if "m" is a minimizer of the mean absolute error with respect to "X". In particular, "m" is a sample median if and only if "m" minimizes the arithmetic mean of the absolute deviations.


title: Median

text: This optimization-based definition of the median is useful in statistical data-analysis, for example, in "k"-medians clustering.


title: Median

text: If the distribution has finite variance, then the distance between the median formula_13 and the mean formula_14 is bounded by one standard deviation.


title: Median

text: This bound was proved by Book and Sher in 1979 for discrete samples, and more generally by Page and Murty in 1982. In a comment on a subsequent proof by O'Cinneide, Mallows in 1991 presented a compact proof that uses Jensen's inequality twice, as follows. Using |·| for the absolute value, we have


title: Median

text: The first and third inequalities come from Jensen's inequality applied to the absolute-value function and the square function, which are each convex. The second inequality comes from the fact that a median minimizes the absolute deviation function formula_16.


title: Median

text: Mallows's proof can be generalized to obtain a multivariate version of the inequality simply by replacing the absolute value with a norm:


title: Median

text: where "m" is a spatial median, that is, a minimizer of the function formula_18 The spatial median is unique when the data-set's dimension is two or more.


title: Median

text: An alternative proof uses the one-sided Chebyshev inequality; it appears in an inequality on location and scale parameters. This formula also follows directly from Cantelli's inequality.


title: Median

text: For the case of unimodal distributions, one can achieve a sharper bound on the distance between the median and the mean:


title: Median

text: Jensen's inequality states that for any random variable "X" with a finite expectation "E"["X"] and for any convex function "f"


title: Median

text: This inequality generalizes to the median as well. We say a function is a C function if, for any "t",


title: Median

text: is a closed interval (allowing the degenerate cases of a single point or an empty set). Every convex function is a C function, but the reverse does not hold. If "f" is a C function, then


title: Median

text: Even though comparison-sorting "n" items requires operations, selection algorithms can compute the th-smallest of items with only operations. This includes the median, which is the th order statistic (or for an even number of samples, the arithmetic mean of the two middle order statistics).


title: Median

text: Selection algorithms still have the downside of requiring memory, that is, they need to have the full sample (or a linear-sized portion of it) in memory. Because this, as well as the linear time requirement, can be prohibitive, several estimation procedures for the median have been developed. A simple one is the median of three rule, which estimates the median as the median of a three-element subsample; this is commonly used as a subroutine in the quicksort sorting algorithm, which uses an estimate of its input's median. A more robust estimator is Tukey's "ninther", which is the median of three rule applied with limited recursion: if is the sample laid out as an array, and


title: Median

text: The "remedian" is an estimator for the median that requires linear time but sub-linear memory, operating in a single pass over the sample.


title: Median

text: The distributions of both the sample mean and the sample median were determined by Laplace. The distribution of the sample median from a population with a density function formula_24 is asymptotically normal with mean formula_25 and variance


title: Median

text: where formula_27 is the median of formula_24 and formula_3 is the sample size. A modern proof follows below. Laplace's result is now understood as a special case of the asymptotic distribution of arbitrary quantiles.


title: Median

text: For normal samples, the density is formula_30, thus for large samples the variance of the median equals formula_31 (See also section #Efficiency below.)


title: Median

text: We take the sample size to be an odd number formula_32 and assume our variable continuous; the formula for the case of discrete variables is given below in . The sample can be summarized as "below median", "at median", and "above median", which corresponds to a trinomial distribution with probabilities formula_33, formula_34 and formula_35. For a continuous variable, the probability of multiple sample values being exactly equal to the median is 0, so one can calculate the density of at the point formula_36 directly from the trinomial distribution:


title: Median

text: Now we introduce the beta function. For integer arguments formula_38 and formula_39, this can be expressed as formula_40. Also, recall that formula_41. Using these relationships and setting both formula_38 and formula_39 equal to formula_44 allows the last expression to be written as


title: Median

text: Hence the density function of the median is a symmetric beta distribution pushed forward by formula_46. Its mean, as we would expect, is 0.5 and its variance is formula_47. By the chain rule, the corresponding variance of the sample median is


title: Median

text: In practice, the functions formula_49 and formula_50 are often not known or assumed. However, they can be estimated from an observed frequency distribution. In this section, we give an example. Consider the following table, representing a sample of 3,800 (discrete-valued) observations:


title: Median

text: Because the observations are discrete-valued, constructing the exact distribution of the median is not an immediate translation of the above expression for formula_51; one may (and typically does) have multiple instances of the median in one's sample. So we must sum over all these possibilities:


title: Median

text: Using these preliminaries, it is possible to investigate the effect of sample size on the standard errors of the mean and median. The observed mean is 3.16, the observed raw median is 3 and the observed interpolated median is 3.174. The following table gives some comparison statistics.


title: Median

text: The expected value of the median falls slightly as sample size increases while, as would be expected, the standard errors of both the median and the mean are proportionate to the inverse square root of the sample size. The asymptotic approximation errs on the side of caution by overestimating the standard error.


title: Median

text: The value of formula_53—the asymptotic value of formula_54 where formula_55 is the population median—has been studied by several authors. The standard "delete one" jackknife method produces inconsistent results. An alternative—the "delete k" method—where formula_56 grows with the sample size has been shown to be asymptotically consistent. This method may be computationally expensive for large data sets. A bootstrap estimate is known to be consistent, but converges very slowly (order of formula_57). Other methods have been proposed but their behavior may differ between large and small samples.


title: Median

text: The efficiency of the sample median, measured as the ratio of the variance of the mean to the variance of the median, depends on the sample size and on the underlying population distribution. For a sample of size formula_58 from the normal distribution, the efficiency for large N is


title: Median

text: In other words, the relative variance of the median will be formula_62, or 57% greater than the variance of the mean – the relative standard error of the median will be formula_63, or 25% greater than the standard error of the mean, formula_64 (see also section #Sampling distribution above.).


title: Median

text: For univariate distributions that are "symmetric" about one median, the Hodges–Lehmann estimator is a robust and highly efficient estimator of the population median.


title: Median

text: If data is represented by a statistical model specifying a particular family of probability distributions, then estimates of the median can be obtained by fitting that family of probability distributions to the data and calculating the theoretical median of the fitted distribution. Pareto interpolation is an application of this when the population is assumed to have a Pareto distribution.


title: Median

text: Previously, this article discussed the univariate median, when the sample or population had one-dimension. When the dimension is two or higher, there are multiple concepts that extend the definition of the univariate median; each such multivariate median agrees with the univariate median when the dimension is exactly one.


title: Median

text: The marginal median is defined for vectors defined with respect to a fixed set of coordinates. A marginal median is defined to be the vector whose components are univariate medians. The marginal median is easy to compute, and its properties were studied by Puri and Sen.


title: Median

text: The geometric median of a discrete set of sample points formula_65 in a Euclidean space is the point minimizing the sum of distances to the sample points.


title: Median

text: In contrast to the marginal median, the geometric median is equivariant with respect to Euclidean similarity transformations such as translations and rotations.


title: Median

text: If the marginal medians for all coordinate systems coincide, then their common location may be termed the "median in all directions". This concept is relevant to voting theory on account of the median voter theorem. When it exists, the median in all directions coincides with the geometric median (at least for discrete distributions).


title: Median

text: When dealing with a discrete variable, it is sometimes useful to regard the observed values as being midpoints of underlying continuous intervals. An example of this is a Likert scale, on which opinions or preferences are expressed on a scale with a set number of possible responses. If the scale consists of the positive integers, an observation of 3 might be regarded as representing the interval from 2.50 to 3.50. It is possible to estimate the median of the underlying variable. If, say, 22% of the observations are of value 2 or below and 55.0% are of 3 or below (so 33% have the value 3), then the median formula_67 is 3 since the median is the smallest value of formula_68 for which formula_69 is greater than a half. But the interpolated median is somewhere between 2.50 and 3.50. First we add half of the interval width formula_70 to the median to get the upper bound of the median interval. Then we subtract that proportion of the interval width which equals the proportion of the 33% which lies above the 50% mark. In other words, we split up the interval width pro rata to the numbers of observations. In this case, the 33% is split into 28% below the median and 5% above it so we subtract 5/33 of the interval width from the upper bound of 3.50 to give an interpolated median of 3.35. More formally, if the values formula_71 are known, the interpolated median can be calculated from


title: Median

text: Alternatively, if in an observed sample there are formula_73 scores above the median category, formula_74 scores in it and formula_75 scores below it then the interpolated median is given by


title: Median

text: For univariate distributions that are "symmetric" about one median, the Hodges–Lehmann estimator is a robust and highly efficient estimator of the population median; for non-symmetric distributions, the Hodges–Lehmann estimator is a robust and highly efficient estimator of the population "pseudo-median", which is the median of a symmetrized distribution and which is close to the population median. The Hodges–Lehmann estimator has been generalized to multivariate distributions.


title: Median

text: The median filter is an important tool of image processing, that can effectively remove any salt and pepper noise from grayscale images.


title: Median

text: In cluster analysis, the k-medians clustering algorithm provides a way of defining clusters, in which the criterion of maximising the distance between cluster-means that is used in k-means clustering, is replaced by maximising the distance between cluster-medians.


title: Median

text: This is a method of robust regression. The idea dates back to Wald in 1940 who suggested dividing a set of bivariate data into two halves depending on the value of the independent parameter formula_2: a left half with values less than the median and a right half with values greater than the median. He suggested taking the means of the dependent formula_78 and independent formula_2 variables of the left and the right halves and estimating the slope of the line joining these two points. The line could then be adjusted to fit the majority of the points in the data set.


title: Median

text: Nair and Shrivastava in 1942 suggested a similar idea but instead advocated dividing the sample into three equal parts before calculating the means of the subsamples. Brown and Mood in 1951 proposed the idea of using the medians of two subsamples rather the means. Tukey combined these ideas and recommended dividing the sample into three equal size subsamples and estimating the line based on the medians of the subsamples.


title: Median

text: Any "mean"-unbiased estimator minimizes the risk (expected loss) with respect to the squared-error loss function, as observed by Gauss. A "median"-unbiased estimator minimizes the risk with respect to the absolute-deviation loss function, as observed by Laplace. Other loss functions are used in statistical theory, particularly in robust statistics.


title: Median

text: Further properties of median-unbiased estimators have been reported. Median-unbiased estimators are invariant under one-to-one transformations.


title: Median

text: There are methods of constructing median-unbiased estimators that are optimal (in a sense analogous to the minimum-variance property for mean-unbiased estimators). Such constructions exist for probability distributions having monotone likelihood-functions. One such procedure is an analogue of the Rao–Blackwell procedure for mean-unbiased estimators: The procedure holds for a smaller class of probability distributions than does the Rao—Blackwell procedure but for a larger class of loss functions.


title: Median

text: Scientific researchers in the ancient near east appear not to have used summary statistics altogether, instead choosing values that offered maximal consistency with a broader theory that integrated a wide variety of phenomena. Within the Mediterranean (and, later, European) scholarly community, statistics like the mean are fundamentally a medieval and early modern development. (The history of the median outside Europe and its predecessors remains relatively unstudied.)


title: Median

text: The idea of the median appeared in the 6th century in the Talmud, in order to fairly analyze divergent appraisals. However, the concept did not spread to the broader scientific community.


title: Median

text: Instead, the closest ancestor of the modern median is the mid-range, invented by Al-Biruni. Transmission of Al-Biruni's work to later scholars is unclear. Al-Biruni applied his technique to assaying metals, but, after he published his work, most assayers still adopted the most unfavorable value from their results, lest they appear to cheat. However, increased navigation at sea during the Age of Discovery meant that ship's navigators increasingly had to attempt to determine latitude in unfavorable weather against hostile shores, leading to renewed interest in summary statistics. Whether rediscovered or independently invented, the mid-range is recommended to nautical navigators in Harriot's "Instructions for Raleigh's Voyage to Guiana, 1595".


title: Median

text: The idea of the median may have first appeared in Edward Wright's 1599 book "Certaine Errors in Navigation" on a section about compass navigation. Wright was reluctant to discard measured values, and may have felt that the median — incorporating a greater proportion of the dataset than the mid-range — was more likely to be correct. However, Wright did not give examples of his technique's use, making it hard to verify that he described the modern notion of median. The median (in the context of probability) certainly appeared in the correspondence of Christiaan Huygens, but as an example of a statistic that was inappropriate for actuarial practice.


title: Median

text: The earliest recommendation of the median dates to 1757, when Roger Joseph Boscovich developed a regression method based on the "L" norm and therefore implicitly on the median. In 1774, Laplace made this desire explicit: he suggested the median be used as the standard estimator of the value of a posterior PDF. The specific criterion was to minimize the expected magnitude of the error; formula_80 where formula_81 is the estimate and formula_82 is the true value. To this end, Laplace determined the distributions of both the sample mean and the sample median in the early 1800s. However, a decade later, Gauss and Legendre developed the least squares method, which minimizes formula_83 to obtain the mean. Within the context of regression, Gauss and Legendre's innovation offers vastly easier computation. Consequently, Laplaces' proposal was generally rejected until the rise of computing devices 150 years later (and is still a relatively uncommon algorithm).


title: Median

text: Antoine Augustin Cournot in 1843 was the first to use the term "median" ("valeur médiane") for the value that divides a probability distribution into two equal halves. Gustav Theodor Fechner used the median ("Centralwerth") in sociological and psychological phenomena. It had earlier been used only in astronomy and related fields. Gustav Fechner popularized the median into the formal analysis of data, although it had been used previously by Laplace, and the median appeared in a textbook by F. Y. Edgeworth. Francis Galton used the English term "median" in 1881, having earlier used the terms "middle-most value" in 1869, and the "medium" in 1880.


title: Median

text: Statisticians encouraged the use of medians intensely throughout the 19th century for its intuitive clarity and ease of manual computation. However, the notion of median does not lend itself to the theory of higher moments as well as the arithmetic mean does, and is much harder to compute by computer. As a result, the median was steadily supplanted as a notion of generic average by the arithmetic mean during the 20th century.


title: Terrifier

text: Terrifier is a 2016 American slasher film written, directed, edited, and produced by Damien Leone. It is a spin-off of the horror anthology film "All Hallows' Eve" (2013). It stars Jenna Kanell, Samantha Scaffidi, Catherine Corcoran, and David Howard Thornton. It follows Tara Heyes (Kanell) on Halloween night, as she becomes the target of a sadistic serial killer known as Art the Clown (Thornton).


title: Terrifier

text: It is the second feature-length film to feature Art, and Leone wrote the film specifically as means of showcasing the character and his practical effects—such as the controversial hacksaw death scene. Leone has expressed regret for leaving the protagonists underdeveloped. Leone filmed "Terrifier" on a low-budget of $35k. It had an Indiegogo campaign, although it ultimately did not reach its goal.


title: Terrifier

text: Mike Gianelli, who portrayed all prior incarnations of Art, retired from acting before production began, and Thornton replaced him as Art. It premiered at the Telluride Horror Show Film Festival in October 2016 before being picked up by Dread Central Presents and Epic Pictures for a limited theatrical release in March 2018. The film received mixed reviews, with praise directed towards the special effects and Thornton's portrayal, while the writing was subject to criticism. The movie quickly became a cult film. A sequel, "Terrifier 2", was released on October 6, 2022.


title: Terrifier

text: A man is watching a small TV where Monica Brown, a talk show host, interviews a severely disfigured woman, the sole survivor of a massacre that took place the previous Halloween. Brown mentions that the body of the killer, known only as "Art the Clown", disappeared from the morgue, suggesting he is still alive. The disfigured woman insists she saw him die. The man furiously kicks the TV and fills a garbage bag with bladed objects. After the interview, Monica talks to her boyfriend on the phone and makes disparaging remarks about the interviewee because of her appearance. The disfigured woman, who'd been eavesdropping, attacks Monica and gouges out her eyes, laughing maniacally.


title: Terrifier

text: On Halloween night, two friends, Tara and Dawn, leave a Halloween party. Art the Clown follows them into a pizzeria. The restaurant owner roughly escorts Art from the premises for smearing his own feces all over the bathroom walls. The girls discover that one of Dawn's car tires has been slashed, and Tara calls her sister Vicky to come pick them up. While waiting, Tara asks a pest control worker, Mike, if she can enter the derelict apartment building he's working in to use the restroom. Once inside, she encounters the Cat Lady, a deluded woman who believes the doll she carries is her infant child. Art kills and mutilates the two pizzeria workers before abducting Dawn.


title: Terrifier

text: Tara encounters Art inside the apartment building. He stabs her with a scalpel and drugs her. Tara awakens bound to a chair, and Art reveals Dawn, suspended upside-down. He forces Tara to watch as he saws Dawn in half with a hacksaw. Tara escapes but Art shoots her to death. The Cat Lady witnesses this and begs Mike to call the police. Mike dismisses her as insane, but Art knocks him unconscious with a hammer. The Cat Lady discovers Art cradling her doll. In a plea for the return of her "child", she tries to show motherly compassion to Art by cradling him.


title: Terrifier

text: Vicky arrives to take Tara and Dawn home but is lured into the basement by Art. There, she discovers Art, who has severely mutilated the Cat Lady and is wearing her scalp and breasts. Mike's co-worker arrives looking for him but is decapitated by Art. Vicky escapes Art but stops to grieve upon finding her sister's corpse. Art then attacks her with a makeshift cat o' nine tails, but Mike arrives suddenly and knocks Art unconscious. The two flee and call 9-1-1 but Art appears and kills Mike. Vicky retreats into a garage, and Art rams through the door with a pickup truck, causing further injury to Vicky. As she lies helplessly, Art begins to eat her face. Police arrive, but Art shoots himself inside his mouth with a pistol before the officers can apprehend him.


title: Terrifier

text: Art's body is taken to a morgue, along with the bodies of his deceased victims. When the medical examiner unzips Art's body bag, Art reanimates and strangles him to death. One year later, Vicky is released from the hospital after rehabilitation from the injuries inflicted by Art; she is revealed to be the severely disfigured woman from the film's opening scene and thus the events of the entire movie took place that previous year.


title: Terrifier

text: The character of Art the Clown first appeared in the 2008 short film "The 9th Circle", which Leone wrote and directed. Leone later wrote and directed a short film titled "Terrifier", which featured Art and was released in 2011. These shorts were incorporated into the 2013 anthology film "All Hallows' Eve", which marked both Art's first feature film appearance and Leone's feature directorial debut.


title: Terrifier

text: In 2015, Leone launched a campaign on the crowdfunding website Indiegogo to finance "Terrifier", a feature-length spin-off of "All Hallows' Eve". After being notified of the Indiegogo campaign, filmmaker Phil Falcone provided the necessary funds for the project in exchange for a producer credit. In "The 9th Circle", the short film "Terrifier", and "All Hallows' Eve", Art was played by Mike Giannelli, but in the feature film "Terrifier", Art was portrayed by David Howard Thornton. Thornton was already familiar with "All Hallows' Eve" when he auditioned for the role of Art in "Terrifier", and was cast in the role after improvising a kill scene in mime.


title: Terrifier

text: "Terrifier" premiered at the Telluride Horror Show Film Festival in 2016. It was later screened at the Horror Channel FrightFest on October 28, 2017, and was subsequently picked up by Dread Central Presents and Epic Pictures for a limited 2018 release.


title: Terrifier

text: "Terrifier" was released on DVD and Blu-ray by Dread Central on March 27, 2018. The release features audio commentary from Damien Leone and David H. Thornton, behind-the-scenes footage, an interview with star Jenna Kanell, deleted scenes, collectible reversible cover art, and several other bonus features.


title: Terrifier

text: On review aggregator Rotten Tomatoes, "Terrifier" holds an approval rating of 56% based on 18 reviews, and an average rating of 5.9/10. While Thornton's portrayal and the special effects were well received by critics, criticism of the film was directed mostly at the dialogue, the acting in certain scenes, and the lack of character development of the protagonists.


title: Terrifier

text: John Higgins ("Starburst") praised the performances of Kanell and Corcoran in that they "are attractive leads and hold the attention." Higgins also praised the film's balance of suspense and gore.


title: Terrifier

text: Anton Bitel of the British Film Institute described the film as a "subtext-free thrill-and-kill ride which openly advertises the sheer senselessness and gratuity of all its on-screen cat-and-mouse deaths by numbers" and "an unapologetically ‘pure’ genre entry, confronting – and amusing – us with all the sinister masked vicariousness of the Halloween spirit." Cody Hamman of Arrow in the Head awarded the film a score of 8 out of 10, calling it "a very simple film, providing 84 minutes of stalking and slashing that occurs largely within the confines of one location. Leone directs the hell out of that simple scenario, though, milking every possible bit of tension from each moment. It's a thrilling, brutal, gory '80s throwback that I recommend checking out, especially if you have a fondness for the same decade of films that this movie obviously holds in high regard." Sol Harris of the magazine "Starburst" gave the film a score of 6 out of 10, writing: "Presented as something of a throwback to horror B-movies of the '80s, "Terrifier" has far more style - both visually and audibly - than the average film of this nature. It's a surprisingly nice looking film for a movie about a clown chopping people into pieces." Jeremy Aspinall of "Radio Times" praised the film, writing "But despite the unsparing gore, there's also plenty of atmosphere and a gnawing tension that's maintained all the way to the sequel-hinting climax." In a thesis by M. Keith Booker, he writes that rather than evolving the slasher film genre in different directions, "Terrifier" acts as a homage to the 1980s films of the subgenre but with better special effects and higher production values. Booker also observes similarities with Dawn's (Corcoran) hacksaw death scene and Freddy Krueger's pursuit of Nancy Thompson in the bathtub scene in "A Nightmare on Elm Street" (1984).


title: Terrifier

text: The film was not without its detractors. Amyana Bartley of "FilmInquiry.com" felt that the film's script lacked both clear protagonists and depth, writing, "Art the Clown has the potential to be a formidable, gruesome, franchise horror character, he just needs more seasoning and cultivation." Felix Vasquez Jr. of Cinema Crazed called it "fairly mediocre slasher fare", stating that the film lacked any creativity and tension while also criticizing its story line. Vasquez concluded his review by stating "As a film "Terrifier" aims high, but feels like a very disposable party favor you'll have forgotten once the credits roll."


title: Terrifier

text: The film received three "Fangoria" Chainsaw Award nominations: Best Limited Release, Best Supporting Actor (Thornton), and Best Makeup FX (Leone).


title: Terrifier

text: In February 2019, Damien Leone stated that a sequel for "Terrifier" was in production, with the other sequel's script having already been written. The film went into production in October 2019 with Fuzz on the Lens Productions as co-producers along with Dark Age Cinema. The film was initially delayed during production of its final days of shooting due to the COVID-19 pandemic, but finally resumed in September 2020 and wrapped shooting in 2021.


title: Terrifier

text: The film debuted at the Arrow Video FrightFest in London, England on August 29, 2022, before receiving a nationwide US cinema release on October 6, 2022, and was released to streaming platforms on November 11, 2022.


title: Clerks III

text: Clerks III is a 2022 American comedy-drama film written, produced, directed, and edited by Kevin Smith and stars Brian O'Halloran, Jeff Anderson, Trevor Fehrman, Austin Zajur, Jason Mewes, Rosario Dawson, and Smith. It serves as a standalone sequel to the 1994 and 2006 "Clerks" films, and is the ninth overall feature film set in the View Askewniverse. In the film, Randal Graves, after surviving a massive heart attack, enlists his friends and fellow clerks Dante Hicks, Elias Grover, and Jay and Silent Bob to make a movie about their lives at the Quick Stop Convenience store that started it all.


title: Clerks III

text: The film had its world premiere on September 4, 2022, in Red Bank, New Jersey, before its release on September 13, by Lionsgate.


title: Clerks III

text: Fifteen years after Dante and Randal bought the Quick Stop, the pair's lives continue much as before, including hockey games on the roof and hanging out with Jay and Silent Bob, who have taken over the former RST Video and turned it into a legal marijuana dispensary. Dante is still grieving over the deaths of Becky and their unborn daughter Grace after they were hit by a drunk driver years prior.


title: Clerks III

text: One day, while Elias and his friend Blockchain are trying to sell their new NFT kites, which contain an image of the Buddy Christ, Randal suffers a severe heart attack. Elias disavows Christianity and decides to become a Satanist after blaming his faith for Randal's heart attack. After the surgery, Randal's doctor urges Dante to keep Randal happy while he recovers, and warns that he is also at risk of the same. Randal decides he has wasted his life and needs a new purpose, declaring he will make a movie about his and Dante's lives at the Quick Stop.


title: Clerks III

text: Becky's spirit appears to Dante, encouraging him to move on. After a harrowing audition process, Randal decides he will not only write and direct but also star in the film as himself, alongside Dante in a supporting role. Rather than cast actors, they decide to cast their friends and actual customers to play themselves. This includes Jay, Silent Bob, and Veronica, who agrees after a sexual encounter with Dante. Dante takes on the producer role and convinces his ex-fiancée Emma to loan the production $30,000, with Dante's half of the Quick Stop as collateral. Silent Bob is made the cinematographer, deciding to shoot the movie in black-and-white, while Elias and Blockchain are hired as production assistants.


title: Clerks III

text: As the shoot for the tentatively-titled "Inconvenience" continues, Dante grows exasperated with Randal's micromanaging and perceived disregard towards him in the script. After Randal tries to recreate the donkey show that took place at Mooby's, Dante has a panic attack and leaves the set as the location reminds him of Becky. Later, a drunk Dante finally confronts Randal for never respecting him or acknowledging his support over the years, and for forcing him to relive the worst moments of his life. Suddenly, he collapses, suffering a heart attack as well. When Randal decides to leave Dante at the hospital to continue work on the movie, Elias lambastes him for his attitude and reveals Dante's deal with Emma. Consumed by guilt, Randal finishes editing the movie and sneaks back into the hospital with Jay and Silent Bob's help. He shows Dante the finished film, which he has re-edited so that Dante is the main character, declaring that the story was always about him. Dante watches the film with the spirit of Becky, touched by his friend's effort, before he peacefully dies.


title: Clerks III

text: Sometime later after Dante's funeral, Emma arrives at the Quick Stop to collect the $30,000 she loaned Dante, or take over the store. Blockchain arrives and excitedly reveals that the NFT kites sold out immediately, making him and Elias millionaires. Elias, who has joined Randal as his new best friend in Dante's place, uses his share of their earnings to pay back Emma. After Elias, Blockchain, Jay, and Silent Bob go outside to fly one of the NFT kites, Randal somberly grieves for Dante at the cash register, unknowingly joined by his spirit. Meanwhile, Jay's daughter Milly inspects the expiration dates on some oat milk.


title: Clerks III

text: During the credits, director Kevin Smith thanks audience members for watching the movie before reading some narration he cut from the final scene, where it is revealed that Randal continued to make movies until the age of 90, all while still working at the Quick Stop.


title: Clerks III

text: Additionally, Fred Armisen, Sarah Michelle Gellar, Freddie Prinze Jr., Melissa Benoist, Chris Wood, Anthony Michael Hall, Danny Trejo, Ralph Garman, and "Impractical Jokers"' Brian Quinn, Sal Vulcano, James Murray, and Joe Gatto all appear as actors auditioning for "Inconvenience". Various customers from the original "Clerks" reprise their roles for "Inconvenience". Other View Askew alumni cameos include John Willyung reprising his role as Cohee Lundin from "Chasing Amy", as well as Scott Mosier and Ethan Suplee each reprising their role of Willam Black from "Clerks" and "Mallrats". Smith's mother Grace, who played the Milk Maid in previous "Clerks" films, appears as an auditioner who declares to whoever wrote the screenplay that their mother should be ashamed of them.


title: Clerks III

text: During press for "Clerks II", Kevin Smith briefly discussed the possibility of a "Clerks III". Stating that "if there's ever gonna be a "Clerks III", it would be somewhere down the road in my 40s or 50s, when it might be interesting to check back in on Dante and Randal. But I don't know about Jay and Bob so much, 'cause at 45, leaning on a wall in front of a convenience store might be a little sad." Smith repeated this sentiment on one of the audio commentary tracks on the "Clerks II" DVD, to which Jeff Anderson jokingly replied, "Oh, don't get me started", referring to Anderson's well known doubts about making "Clerks II" when first approached by Smith.


title: Clerks III

text: On March 29, 2012, Smith expressed his interest in producing "Clerks III" as a Broadway play after seeing the Theresa Rebeck comedy "Seminar" starring Alan Rickman, with whom Smith had previously worked on "Dogma".


title: Clerks III

text: On December 10, 2012, Smith released a special Hollywood Babble-On episode, "Hollywood Babble-On #000: GIANT SIZED ANNUAL # 1: CLERKS III, AUDIENCE 0", in which he revealed greater details on his plans for "Clerks III". Smith stated that an ongoing audit over residuals from "Clerks II" with The Weinstein Company was causing a delay in several key "Clerks III" cast and crew members, including Anderson and Scott Mosier, from coming on board until the audit was resolved. Smith also revealed that he would like to crowdsource "Clerks III", either through Kickstarter or Indiegogo, with contributors receiving anything from DVDs, posters, and even roles as extras in the film. On June 5, 2013, he changed his mind on crowdsourcing, stating "I've got access to money. And worst-case scenario, I can put up my house."


title: Clerks III

text: Smith worked on a script for "Clerks III" from March to May 2013, stating when he completed it that it was ""The Empire Strikes Back"" of the series. This script featured a story about Randal having a nervous breakdown after the Quick Stop is destroyed during Hurricane Sandy, and trying to manage it by getting in line for a film called "Ranger Danger" a year before it opens. Randall would have gained a small cult following and set up his own miniature Quick Stop, only for there to be a shooting at the theater. In July 2013, Jason Mewes stated that they were now just waiting to hear back from The Weinstein Company about funding.


title: Clerks III

text: On September 26, 2014, Smith stated on his "Hollywood Babble-On" podcast, that he was glad that he made "Tusk", so he could go on to make "Clerks III":


title: Clerks III

text: There were plans to start shooting "Clerks III" in May 2015, but these were put on hold to film another sequel, "Mallrats 2". However, by June 2016 the plans for a "Mallrats" sequel had been turned into plans for a "Mallrats" TV series.


title: Clerks III

text: The initial plans to film "Clerks III" came to a halt in 2017, when Smith announced that one of the four leads, whom he later revealed to be Anderson, opted out of reprising his role despite a completed script. At the time, Smith doubted the film would ever be made. In 2018, Smith suffered a near-fatal heart attack after one of his comedy shows. This experience inspired him to rewrite the "Clerks III" script from scratch, doing away with the original planned storyline. Smith later reflected that the original script strayed too far from the original "Clerks", noting it was written by "a guy who didn't know a thing about death" and he was ultimately glad it was never made. Smith later reused the original opening scene, featuring Jay and Silent Bob getting arrested (alongside Dante and Randal in the original "Clerks III" script) for "Jay and Silent Bob Reboot".


title: Clerks III

text: In July 2019, Smith announced that he would do a live reading of the original, now-aborted "Clerks III" script at the First Avenue Playhouse in Atlantic Highlands, New Jersey. The reading was held on August 3, 2019. That same month, Smith revealed at San Diego Comic Con that he was writing a new script for "Clerks III" and promised to make the film.


title: Clerks III

text: On October 1, 2019, Smith confirmed on Instagram that "Clerks III" was happening and that Jeff Anderson agreed to reprise his role as Randal. The new script will follow Randal, after surviving a heart attack, and Dante making a movie about their lives at the store, a plot initially conceived as a film adaptation of "" titled "Clerks: Sell Out".


title: Clerks III

text: In January 2021, Smith announced the script was complete. In July 2021, Lionsgate obtained the rights to produce and distribute the film, while also confirming Rosario Dawson and Trevor Fehrman would reprise their roles of Becky Scott and Elias Grover from "Clerks II" respectively.


title: Clerks III

text: Principal photography began on August 2, 2021, in Red Bank, New Jersey. Filming wrapped on August 31, 2021.


title: Clerks III

text: Post-production was completed on February 13, 2022, with Smith announcing a trailer was slated to debut in May. On May 28, 2022, Smith stated he viewed the first cut of the trailer and was set to release it during San Diego Comic-Con 2022, which would run from July 21–24. On June 29, 2022, Smith announced the trailer would instead drop online on July 6.


title: Clerks III

text: The film is dedicated to Lisa Spoonauer who played Caitlin Bree in the original "Clerks" and "". Spoonauer passed away in 2017.


title: Clerks III

text: "Clerks III" had its world premiere on August 24, 2022 in Los Angeles, followed by a roadshow tour that began September 4, 2022, in Red Bank, New Jersey. It was released on September 13, 2022 by Lionsgate and Fathom Events. The film was previously scheduled for release in July 2022.


title: Clerks III

text: The film released digitally on October 14, 2022, and was released on 4K Ultra HD, Blu-ray and DVD on December 6, 2022.


title: Clerks III

text: On review aggregator website Rotten Tomatoes, the film has an approval rating of 63% based on 124 reviews, with an average rating of 6.1/10. The site's critics consensus reads: ""Clerks III" isn't even supposed to be here today -- but this surprisingly emotional return to the Quick Stop wraps up the trilogy in fan-pleasing fashion." Metacritic assigned the film a weighted average score of 50 out of 100, based on 25 critics, indicating "mixed or average reviews".


title: Function (mathematics)

text: In mathematics, a function from a set to a set assigns to each element of exactly one element of . The set is called the domain of the function and the set is called the codomain of the function.


title: Function (mathematics)

text: The earliest known approach to the notion of function can be traced back to works of Persian mathematicians Al-Biruni and Sharaf al-Din al-Tusi. Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a "function" of time. Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable (that is, they had a high degree of regularity). The concept of a function was formalized at the end of the 19th century in terms of set theory, and this greatly enlarged the domains of application of the concept.


title: Function (mathematics)

text: A function is most often denoted by letters such as , and , and the value of a function at an element of its domain is denoted by ; the numerical value resulting from the "function evaluation" at a particular input value is denoted by replacing with this value; for example, the value of at is denoted by . When the function is not named and is represented by an expression , the value of the function at, say, may be denoted by . For example, the value at of the function that maps to formula_1 may be denoted by formula_2 (which results in


title: Function (mathematics)

text: A function is uniquely represented by the set of all pairs , called the "graph of the function", a popular means of illustrating the function. When the domain and the codomain are sets of real numbers, each such pair may be thought of as the Cartesian coordinates of a point in the plane.


title: Function (mathematics)

text: Functions are widely used in science, engineering, and in most fields of mathematics. It has been said that functions are "the central objects of investigation" in most fields of mathematics.


title: Function (mathematics)

text: A function from a set to a set is an assignment of an element of to each element of . The set is called the domain of the function and the set is called the codomain of the function.


title: Function (mathematics)

text: A function, its domain, and its codomain, are declared by the notation , and the value of a function at an element of , denoted by , is called the "image" of under , or the "value" of applied to the "argument" .


title: Function (mathematics)

text: Functions are also called "maps" or "mappings", though some authors make some distinction between "maps" and "functions" (see ).


title: Function (mathematics)

text: Two functions and are equal if their domain and codomain sets are the same and their output values agree on the whole domain. More formally, given and , we have if and only if for all .


title: Function (mathematics)

text: The domain and codomain are not always explicitly given when a function is defined, and, without some (possibly difficult) computation, one might only know that the domain is contained in a larger set. Typically, this occurs in mathematical analysis, where "a function often refers to a function that may have a proper subset of as domain. For example, a "function from the reals to the reals" may refer to a real-valued function of a real variable. However, a "function from the reals to the reals" does not mean that the domain of the function is the whole set of the real numbers, but only that the domain is a set of real numbers that contains a non-empty open interval. Such a function is then called a partial function. For example, if is a function that has the real numbers as domain and codomain, then a function mapping the value to the value is a function from the reals to the reals, whose domain is the set of the reals , such that .


title: Function (mathematics)

text: Any subset of the Cartesian product of two sets and defines a binary relation between these two sets. It is immediate that an arbitrary relation may contain pairs that violate the necessary conditions for a function given above.


title: Function (mathematics)

text: A partial function is a binary relation that is univalent, and a function is a binary relation that is univalent and total.


title: Function (mathematics)

text: Various properties of functions and function composition may be reformulated in the language of relations. For example, a function is injective if the converse relation is univalent, where the converse relation is defined as


title: Function (mathematics)

text: This notation is the same as the notation for the Cartesian product of a family of copies of formula_6 indexed by formula_5:


title: Function (mathematics)

text: The identity of these two notations is motivated by the fact that a function formula_13 can be identified with the element of the Cartesian product such that the component of index formula_14 is formula_15.


title: Function (mathematics)

text: When formula_6 has two elements, formula_17 is commonly denoted formula_18 and called the powerset of . It can be identified with the set of all subsets of formula_5, through the one-to-one correspondence that associates to each subset formula_20 the function formula_13 such that formula_22 if formula_23 and formula_24 otherwise.


title: Function (mathematics)

text: There are various standard ways for denoting functions. The most commonly used notation is functional notation, which is the first notation described below.


title: Function (mathematics)

text: In functional notation, the function is immediately given a name, such as , and its definition is given by what does to the explicit argument , using a formula in terms of . For example, the function which takes a real number as input and outputs that number plus 1 is denoted by


title: Function (mathematics)

text: If a function is defined in this notation, its domain and codomain are implicitly taken to both be formula_26, the set of real numbers. If the formula cannot be evaluated at all real numbers, then the domain is implicitly taken to be the maximal subset of formula_26 on which the formula can be evaluated; see Domain of a function.


title: Function (mathematics)

text: In this example, the function takes a real number as input, squares it, then adds 1 to the result, then takes the sine of the result, and returns the final result as the output.


title: Function (mathematics)

text: When the symbol denoting the function consists of several characters and no ambiguity may arise, the parentheses of functional notation might be omitted. For example, it is common to write instead of .


title: Function (mathematics)

text: Functional notation was first used by Leonhard Euler in 1734. Some widely used functions are represented by a symbol consisting of several letters (usually two or three, generally an abbreviation of their name). In this case, a roman type is customarily used instead, such as "" for the sine function, in contrast to italic font for single-letter symbols.


title: Function (mathematics)

text: When using this notation, one often encounters the abuse of notation whereby the notation can refer to the value of at , or to the function itself. If the variable was previously declared, then the notation unambiguously means the value of at . Otherwise, it is useful to understand the notation as being both simultaneously; this allows one to denote composition of two functions and in a succinct manner by the notation .


title: Function (mathematics)

text: However, distinguishing and can become important in cases where functions themselves serve as inputs for other functions. (A function taking another function as an input is termed a "functional".) Other approaches of notating functions, detailed below, avoid this problem but are less commonly used.


title: Function (mathematics)

text: Arrow notation defines the rule of a function inline, without requiring a name to be given to the function. For example, formula_29 is the function which takes a real number as input and outputs that number plus 1. Again a domain and codomain of formula_26 is implied.


title: Function (mathematics)

text: As a common application of the arrow notation, suppose formula_32 is a function in two variables, and we want to refer to a partially applied function formula_33 produced by fixing the second argument to the value without introducing a new function name. The map in question could be denoted formula_34 using the arrow notation. The expression formula_34 (read: "the map taking to ") represents this new function with just one argument, whereas the expression refers to the value of the function at the


title: Function (mathematics)

text: Index notation is often used instead of functional notation. That is, instead of writing , one writes formula_36


title: Function (mathematics)

text: This is typically the case for functions whose domain is the set of the natural numbers. Such a function is called a sequence, and, in this case the element formula_37 is called the th element of the sequence.


title: Function (mathematics)

text: The index notation is also often used for distinguishing some variables called parameters from the "true variables". In fact, parameters are specific variables that are considered as being fixed during the study of a problem. For example, the map formula_38 (see above) would be denoted formula_39 using index notation, if we define the collection of maps formula_39 by the formula formula_41 for all formula_42.


title: Function (mathematics)

text: the symbol does not represent any value, it is simply a placeholder meaning that, if is replaced by any value on the left of the arrow, it should be replaced by the same value on the right of the arrow. Therefore, may be replaced by any symbol, often an interpunct "". This may be useful for distinguishing the function from its value at .


title: Function (mathematics)

text: For example, formula_44 may stand for the function formula_45, and formula_46 may stand for a function defined by an integral with variable upper bound: formula_47.


title: Function (mathematics)

text: There are other, specialized notations for functions in sub-disciplines of mathematics. For example, in linear algebra and functional analysis, linear forms and the vectors they act upon are denoted using a dual pair to show the underlying duality. This is similar to the use of bra–ket notation in quantum mechanics. In logic and the theory of computation, the function notation of lambda calculus is used to explicitly express the basic notions of function abstraction and application. In category theory and homological algebra, networks of functions are described in terms of how they and their compositions commute with each other using commutative diagrams that extend and generalize the arrow notation for functions described above.


title: Function (mathematics)

text: A function is often also called a map or a mapping, but some authors make a distinction between the term "map" and "function". For example, the term "map" is often reserved for a "function" with some sort of special structure (e.g. maps of manifolds). In particular "map" is often used in place of "homomorphism" for the sake of succinctness (e.g., linear map or "map from to " instead of "group homomorphism from to "). Some authors reserve the word "mapping" for the case where the structure of the codomain belongs explicitly to the definition of the function.


title: Function (mathematics)

text: Some authors, such as Serge Lang, use "function" only to refer to maps for which the codomain is a subset of the real or complex numbers, and use the term "mapping" for more general functions.


title: Function (mathematics)

text: In the theory of dynamical systems, a map denotes an evolution function used to create discrete dynamical systems. See also Poincaré map.


title: Function (mathematics)

text: Whichever definition of "map" is used, related terms like "domain", "codomain", "injective", "continuous" have the same meaning as for a function.


title: Function (mathematics)

text: Given a function formula_13, by definition, to each element formula_14 of the domain of the function formula_13, there is a unique element associated to it, the value formula_15 of formula_13 at formula_14. There are several ways to specify or describe how formula_14 is related to formula_15, both explicitly and implicitly. Sometimes, a theorem or an axiom asserts the existence of a function having some properties, without describing it more precisely. Often, the specification or description is referred to as the definition of the function formula_13.


title: Function (mathematics)

text: On a finite set, a function may be defined by listing the elements of the codomain that are associated to the elements of the domain. For example, if formula_57, then one can define a function formula_58 by formula_59


title: Function (mathematics)

text: Functions are often defined by a formula that describes a combination of arithmetic operations and previously defined functions; such a formula allows computing the value of the function from the value of any element of the domain.


title: Function (mathematics)

text: For example, in the above example, formula_13 can be defined by the formula formula_61, for formula_62.


title: Function (mathematics)

text: When a function is defined this way, the determination of its domain is sometimes difficult. If the formula that defines the function contains divisions, the values of the variable for which a denominator is zero must be excluded from the domain; thus, for a complicated function, the determination of the domain passes through the computation of the zeros of auxiliary functions. Similarly, if square roots occur in the definition of a function from formula_63 to formula_64 the domain is included in the set of the values of the variable for which the arguments of the square roots are nonnegative.


title: Function (mathematics)

text: For example, formula_65 defines a function formula_66 whose domain is formula_64 because formula_68 is always positive if is a real number. On the other hand, formula_69 defines a function from the reals to the reals whose domain is reduced to the interval . (In old texts, such a domain was called the "domain of definition" of the function.)


title: Function (mathematics)

text: A function formula_75 with domain and codomain , is bijective, if for every in , there is one and only one element in such that . In this case, the inverse function of is the function formula_76 that maps formula_77 to the element formula_78 such that . For example, the natural logarithm is a bijective function from the positive real numbers to the real numbers. It thus has an inverse, called the exponential function, that maps the real numbers onto the positive numbers.


title: Function (mathematics)

text: If a function formula_79 is not bijective, it may occur that one can select subsets formula_80 and formula_81 such that the restriction of to is a bijection from to , and has thus an inverse. The inverse trigonometric functions are defined this way. For example, the cosine function induces, by restriction, a bijection from the interval onto the interval , and its inverse function, called arccosine, maps onto . The other inverse trigonometric functions are defined similarly.


title: Function (mathematics)

text: More generally, given a binary relation between two sets and , let be a subset of such that, for every formula_82 there is some formula_77 such that . If one has a criterion allowing selecting such an for every formula_82 this defines a function formula_85 called an implicit function, because it is implicitly defined by the relation .


title: Function (mathematics)

text: For example, the equation of the unit circle formula_86 defines a relation on real numbers. If there are two possible values of , one positive and one negative. For , these two values become both equal to 0. Otherwise, there is no possible value of . This means that the equation defines two implicit functions with domain and respective codomains and .


title: Function (mathematics)

text: In this example, the equation can be solved in , giving formula_87 but, in more complicated examples, this is impossible. For example, the relation formula_88 defines as an implicit function of , called the Bring radical, which has formula_89 as domain and range. The Bring radical cannot be expressed in terms of the four arithmetic operations and th roots.


title: Function (mathematics)

text: The implicit function theorem provides mild differentiability conditions for existence and uniqueness of an implicit function in the neighborhood of a point.


title: Function (mathematics)

text: Many functions can be defined as the antiderivative of another function. This is the case of the natural logarithm, which is the antiderivative of that is 0 for . Another common example is the error function.


title: Function (mathematics)

text: More generally, many functions, including most special functions, can be defined as solutions of differential equations. The simplest example is probably the exponential function, which can be defined as the unique function that is equal to its derivative and takes the value 1 for .


title: Function (mathematics)

text: Power series can be used to define functions on the domain in which they converge. For example, the exponential function is given by formula_90. However, as the coefficients of a series are quite arbitrary, a function that is the sum of a convergent series is generally defined otherwise, and the sequence of the coefficients is the result of some computation based on another definition. Then, the power series can be used to enlarge the domain of the function. Typically, if a function for a real variable is the sum of its Taylor series in some interval, this power series allows immediately enlarging the domain to a subset of the complex numbers, the disc of convergence of the series. Then analytic continuation allows enlarging further the domain for including almost the whole complex plane. This process is the method that is generally used for defining the logarithm, the exponential and the trigonometric functions of a complex number.


title: Function (mathematics)

text: Functions whose domain are the nonnegative integers, known as sequences, are often defined by recurrence relations.


title: Function (mathematics)

text: The factorial function on the nonnegative integers (formula_91) is a basic example, as it can be defined by the recurrence relation


title: Function (mathematics)

text: A graph is commonly used to give an intuitive picture of a function. As an example of how a graph helps to understand a function, it is easy to see from its graph whether a function is increasing or decreasing. Some functions may also be represented by bar charts.


title: Function (mathematics)

text: In the frequent case where and are subsets of the real numbers (or may be identified with such subsets, e.g. intervals), an element formula_96 may be identified with a point having coordinates in a 2-dimensional coordinate system, e.g. the Cartesian plane. Parts of this may create a plot that represents (parts of) the function. The use of plots is so ubiquitous that they too are called the "graph of the function". Graphic representations of functions are also possible in other coordinate systems. For example, the graph of the square function


title: Function (mathematics)

text: consisting of all points with coordinates formula_98 for formula_99 yields, when depicted in Cartesian coordinates, the well known parabola. If the same quadratic function formula_97 with the same formal graph, consisting of pairs of numbers, is plotted instead in polar coordinates formula_101 the plot obtained is Fermat's spiral.


title: Function (mathematics)

text: A function can be represented as a table of values. If the domain of a function is finite, then the function can be completely specified in this way. For example, the multiplication function formula_102 defined as formula_103 can be represented by the familiar multiplication table


title: Function (mathematics)

text: On the other hand, if a function's domain is continuous, a table can give the values of the function at specific values of the domain. If an intermediate value is needed, interpolation can be used to estimate the value of the function. For example, a portion of a table for the sine function might be given as follows, with values rounded to 6 decimal places:


title: Function (mathematics)

text: Before the advent of handheld calculators and personal computers, such tables were often compiled and published for functions such as logarithms and trigonometric functions.


title: Function (mathematics)

text: Bar charts are often used for representing functions whose domain is a finite set, the natural numbers, or the integers. In this case, an element of the domain is represented by an interval of the -axis, and the corresponding value of the function, , is represented by a rectangle whose base is the interval corresponding to and whose height is (possibly negative, in which case the bar extends below the -axis).


title: Function (mathematics)

text: This section describes general properties of functions, that are independent of specific properties of the domain and the codomain.


title: Function (mathematics)

text: Given two functions formula_79 and formula_110 such that the domain of is the codomain of , their "composition" is the function formula_111 defined by


title: Function (mathematics)

text: That is, the value of formula_113 is obtained by first applying to to obtain and then applying to the result to obtain . In the notation the function that is applied first is always written on the right.


title: Function (mathematics)

text: The composition formula_114 is an operation on functions that is defined only if the codomain of the first function is the domain of the second one. Even when both formula_113 and formula_116 satisfy these conditions, the composition is not necessarily commutative, that is, the functions formula_113 and formula_118 need not be equal, but may deliver different values for the same argument. For example, let and , then formula_119 and formula_120 agree just for formula_121


title: Function (mathematics)

text: The function composition is associative in the sense that, if one of formula_122 and formula_123 is defined, then the other is also defined, and they are equal. Thus, one writes


title: Function (mathematics)

text: The identity functions formula_125 and formula_126 are respectively a right identity and a left identity for functions from to . That is, if is a function with domain , and codomain , one has


title: Function (mathematics)

text: Let formula_128 The "image" under of an element of the domain is . If is any subset of , then the "image" of under , denoted , is the subset of the codomain consisting of all images of elements of , that is,


title: Function (mathematics)

text: The "image" of is the image of the whole domain, that is, . It is also called the range of , although the term "range" may also refer to the codomain.


title: Function (mathematics)

text: On the other hand, the "inverse image" or "preimage" under of an element of the codomain is the set of all elements of the domain whose images under equal . In symbols, the preimage of is denoted by formula_130 and is given by the equation


title: Function (mathematics)

text: Likewise, the preimage of a subset of the codomain is the set of the preimages of the elements of , that is, it is the subset of the domain consisting of all elements of whose images belong to . It is denoted by formula_132 and is given by the equation


title: Function (mathematics)

text: By definition of a function, the image of an element of the domain is always a single element of the codomain. However, the preimage formula_130 of an element of the codomain may be empty or contain any number of elements. For example, if is the function from the integers to themselves that maps every integer to 0, then formula_137.


title: Function (mathematics)

text: If formula_79 is a function, and are subsets of , and and are subsets of , then one has the following properties:


title: Function (mathematics)

text: The preimage by of an element of the codomain is sometimes called, in some contexts, the fiber of under .


title: Function (mathematics)

text: If a function has an inverse (see below), this inverse is denoted formula_145 In this case formula_146 may denote either the image by formula_147 or the preimage by of . This is not a problem, as these sets are equal. The notation formula_148 and formula_146 may be ambiguous in the case of sets that contain some subsets as elements, such as formula_150 In this case, some care may be needed, for example, by using square brackets formula_151 for images and preimages of subsets and ordinary parentheses for images and preimages of elements.


title: Function (mathematics)

text: The function is "injective" (or "one-to-one", or is an "injection") if for any two different elements and of . Equivalently, is injective if and only if, for any formula_153 the preimage formula_130 contains at most one element. An empty function is always injective. If is not the empty set, then is injective if and only if there exists a function formula_155 such that formula_156 that is, if has a left inverse. "Proof": If is injective, for defining , one chooses an element formula_157 in (which exists as is supposed to be nonempty), and one defines by formula_158 if formula_159 and formula_160 if formula_161 Conversely, if formula_156 and formula_163 then formula_164 and thus formula_165


title: Function (mathematics)

text: The function is "surjective" (or "onto", or is a "surjection") if its range formula_166 equals its codomain formula_6, that is, if, for each element formula_168 of the codomain, there exists some element formula_14 of the domain such that formula_170 (in other words, the preimage formula_130 of every formula_77 is nonempty). If, as usual in modern mathematics, the axiom of choice is assumed, then is surjective if and only if there exists a function formula_155 such that formula_174 that is, if has a right inverse. The axiom of choice is needed, because, if is surjective, one defines by formula_175 where formula_14 is an "arbitrarily chosen" element of formula_177


title: Function (mathematics)

text: The function is "bijective" (or is a "bijection" or a "one-to-one correspondence") if it is both injective and surjective. That is, is bijective if, for any formula_153 the preimage formula_130 contains exactly one element. The function is bijective if and only if it admits an inverse function, that is, a function formula_155 such that formula_181 and formula_182 (Contrarily to the case of surjections, this does not require the axiom of choice; the proof is straightforward).


title: Function (mathematics)

text: Every function formula_79 may be factorized as the composition formula_184 of a surjection followed by an injection, where is the canonical surjection of onto and is the canonical injection of into . This is the "canonical factorization" of .


title: Function (mathematics)

text: "One-to-one" and "onto" are terms that were more common in the older English language literature; "injective", "surjective", and "bijective" were originally coined as French words in the second quarter of the 20th century by the Bourbaki group and imported into English. As a word of caution, "a one-to-one function" is one that is injective, while a "one-to-one correspondence" refers to a bijective function. Also, the statement " maps "onto" " differs from " maps "into" ", in that the former implies that is surjective, while the latter makes no assertion about the nature of . In a complicated reasoning, the one letter difference can easily be missed. Due to the confusing nature of this older terminology, these terms have declined in popularity relative to the Bourbakian terms, which have also the advantage of being more symmetrical.


title: Function (mathematics)

text: If formula_185 is a function and "S" is a subset of "X", then the "restriction" of formula_13 to "S", denoted formula_187, is the function from "S" to "Y" defined by


title: Function (mathematics)

text: for all "x" in "S". Restrictions can be used to define partial inverse functions: if there is a subset "S" of the domain of a function formula_13 such that formula_187 is injective, then the canonical surjection of formula_187 onto its image formula_192 is a bijection, and thus has an inverse function from formula_193 to "S". One application is the definition of inverse trigonometric functions. For example, the cosine function is injective when restricted to the interval . The image of this restriction is the interval , and thus the restriction has an inverse function from to , which is called arccosine and is denoted .


title: Function (mathematics)

text: Function restriction may also be used for "gluing" functions together. Let formula_194 be the decomposition of as a union of subsets, and suppose that a function formula_195 is defined on each formula_196 such that for each pair formula_197 of indices, the restrictions of formula_198 and formula_199 to formula_200 are equal. Then this defines a unique function formula_185 such that formula_202 for all . This is the way that functions on manifolds are defined.


title: Function (mathematics)

text: An "extension" of a function is a function such that is a restriction of . A typical use of this concept is the process of analytic continuation, that allows extending functions whose domain is a small part of the complex plane to functions whose domain is almost the whole complex plane.


title: Function (mathematics)

text: Here is another classical example of a function extension that is encountered when studying homographies of the real line. A "homography" is a function formula_203 such that . Its domain is the set of all real numbers different from formula_204 and its image is the set of all real numbers different from formula_205 If one extends the real line to the projectively extended real line by including , one may extend to a bijection from the extended real line to itself by setting formula_206 and formula_207.


title: Function (mathematics)

text: A multivariate function, or function of several variables is a function that depends on several arguments. Such functions are commonly encountered. For example, the position of a car on a road is a function of the time travelled and its average speed.


title: Function (mathematics)

text: For example, multiplication of integers is a function of two variables, or bivariate function, whose domain is the set of all pairs (2-tuples) of integers, and whose codomain is the set of integers. The same is true for every binary operation. More generally, every mathematical operation is defined as a multivariate function.


title: Function (mathematics)

text: The Cartesian product formula_208 of sets formula_209 is the set of all -tuples formula_210 such that formula_211 for every with formula_212. Therefore, a function of variables is a function


title: Function (mathematics)

text: When using function notation, one usually omits the parentheses surrounding tuples, writing formula_215 instead of formula_216


title: Function (mathematics)

text: In the case where all the formula_217 are equal to the set formula_26 of real numbers, one has a function of several real variables. If the formula_217 are equal to the set formula_220 of complex numbers, one has a function of several complex variables.


title: Function (mathematics)

text: It is common to also consider functions whose codomain is a product of sets. For example, Euclidean division maps every pair of integers with to a pair of integers called the "quotient" and the "remainder":


title: Function (mathematics)

text: The codomain may also be a vector space. In this case, one talks of a vector-valued function. If the domain is contained in a Euclidean space, or more generally a manifold, a vector-valued function is often called a vector field.


title: Function (mathematics)

text: The idea of function, starting in the 17th century, was fundamental to the new infinitesimal calculus. At that time, only real-valued functions of a real variable were considered, and all functions were assumed to be smooth. But the definition was soon extended to functions of several variables and to functions of a complex variable. In the second half of the 19th century, the mathematically rigorous definition of a function was introduced, and functions with arbitrary domains and codomains were defined.


title: Function (mathematics)

text: Functions are now used throughout all areas of mathematics. In introductory calculus, when the word "function" is used without qualification, it means a real-valued function of a single real variable. The more general definition of a function is usually introduced to second or third year college students with STEM majors, and in their senior year they are introduced to calculus in a larger, more rigorous setting in courses such as real analysis and complex analysis.


title: Function (mathematics)

text: A "real function" is a real-valued function of a real variable, that is, a function whose codomain is the field of real numbers and whose domain is a set of real numbers that contains an interval. In this section, these functions are simply called "functions".


title: Function (mathematics)

text: The functions that are most commonly considered in mathematics and its applications have some regularity, that is they are continuous, differentiable, and even analytic. This regularity insures that these functions can be visualized by their graphs. In this section, all functions are differentiable in some interval.


title: Function (mathematics)

text: Functions enjoy pointwise operations, that is, if and are functions, their sum, difference and product are functions defined by


title: Function (mathematics)

text: The domains of the resulting functions are the intersection of the domains of and . The quotient of two functions is defined similarly by


title: Function (mathematics)

text: but the domain of the resulting function is obtained by removing the zeros of from the intersection of the domains of and .


title: Function (mathematics)

text: The polynomial functions are defined by polynomials, and their domain is the whole set of real numbers. They include constant functions, linear functions and quadratic functions. Rational functions are quotients of two polynomial functions, and their domain is the real numbers with a finite number of them removed to avoid division by zero. The simplest rational function is the function formula_224 whose graph is a hyperbola, and whose domain is the whole real line except for 0.


title: Function (mathematics)

text: The derivative of a real differentiable function is a real function. An antiderivative of a continuous real function is a real function that has the original function as a derivative. For example, the function formula_225 is continuous, and even differentiable, on the positive real numbers. Thus one antiderivative, which takes the value zero for , is a differentiable function called the natural logarithm.


title: Function (mathematics)

text: A real function is monotonic in an interval if the sign of formula_226 does not depend of the choice of and in the interval. If the function is differentiable in the interval, it is monotonic if the sign of the derivative is constant in the interval. If a real function is monotonic in an interval , it has an inverse function, which is a real function with domain and image . This is how inverse trigonometric functions are defined in terms of trigonometric functions, where the trigonometric functions are monotonic. Another example: the natural logarithm is monotonic on the positive real numbers, and its image is the whole real line; therefore it has an inverse function that is a bijection between the real numbers and the positive real numbers. This inverse is the exponential function.


title: Function (mathematics)

text: Many other real functions are defined either by the implicit function theorem (the inverse function is a particular instance) or as solutions of differential equations. For example, the sine and the cosine functions are the solutions of the linear differential equation


title: Function (mathematics)

text: When the elements of the codomain of a function are vectors, the function is said to be a vector-valued function. These functions are particularly useful in applications, for example modeling physical properties. For example, the function that associates to each point of a fluid its velocity vector is a vector-valued function.


title: Function (mathematics)

text: Some vector-valued functions are defined on a subset of formula_229 or other spaces that share geometric or topological properties of formula_229, such as manifolds. These vector-valued functions are given the name "vector fields".


title: Function (mathematics)

text: In mathematical analysis, and more specifically in functional analysis, a function space is a set of scalar-valued or vector-valued functions, which share a specific property and form a topological vector space. For example, the real smooth functions with a compact support (that is, they are zero outside some compact set) form a function space that is at the basis of the theory of distributions.


title: Function (mathematics)

text: Function spaces play a fundamental role in advanced mathematical analysis, by allowing the use of their algebraic and topological properties for studying properties of functions. For example, all theorems of existence and uniqueness of solutions of ordinary or partial differential equations result of the study of function spaces.


title: Function (mathematics)

text: Several methods for specifying functions of real or complex variables start from a local definition of the function at a point or on a neighbourhood of a point, and then extend by continuity the function to a much larger domain. Frequently, for a starting point formula_231 there are several possible starting values for the function.


title: Function (mathematics)

text: For example, in defining the square root as the inverse function of the square function, for any positive real number formula_231 there are two choices for the value of the square root, one of which is positive and denoted formula_233 and another which is negative and denoted formula_234 These choices define two continuous functions, both having the nonnegative real numbers as a domain, and having either the nonnegative or the nonpositive real numbers as images. When looking at the graphs of these functions, one can see that, together, they form a single smooth curve. It is therefore often useful to consider these two square root functions as a single function that has two values for positive , one value for 0 and no value for negative .


title: Function (mathematics)

text: In the preceding example, one choice, the positive square root, is more natural than the other. This is not the case in general. For example, let consider the implicit function that maps to a root of formula_235 (see the figure on the right). For one may choose either formula_236 for . By the implicit function theorem, each choice defines a function; for the first one, the (maximal) domain is the interval and the image is ; for the second one, the domain is and the image is ; for the last one, the domain is and the image is . As the three graphs together form a smooth curve, and there is no reason for preferring one choice, these three functions are often considered as a single "multi-valued function" of that has three values for , and only one value for and .


title: Function (mathematics)

text: Usefulness of the concept of multi-valued functions is clearer when considering complex functions, typically analytic functions. The domain to which a complex function may be extended by analytic continuation generally consists of almost the whole complex plane. However, when extending the domain through two different paths, one often gets different values. For example, when extending the domain of the square root function, along a path of complex numbers with positive imaginary parts, one gets for the square root of −1; while, when extending through complex numbers with negative imaginary parts, one gets . There are generally two ways of solving the problem. One may define a function that is not continuous along some curve, called a branch cut. Such a function is called the principal value of the function. The other way is to consider that one has a "multi-valued function", which is analytic everywhere except for isolated singularities, but whose value may "jump" if one follows a closed loop around a singularity. This jump is called the monodromy.


title: Function (mathematics)

text: The definition of a function that is given in this article requires the concept of set, since the domain and the codomain of a function must be a set. This is not a problem in usual mathematics, as it is generally not difficult to consider only functions whose domain and codomain are sets, which are well defined, even if the domain is not explicitly defined. However, it is sometimes useful to consider more general functions.


title: Function (mathematics)

text: For example, the singleton set may be considered as a function formula_237 Its domain would include all sets, and therefore would not be a set. In usual mathematics, one avoids this kind of problem by specifying a domain, which means that one has many singleton functions. However, when establishing foundations of mathematics, one may have to use functions whose domain, codomain or both are not specified, and some authors, often logicians, give precise definition for these weakly specified functions.


title: Function (mathematics)

text: These generalized functions may be critical in the development of a formalization of the foundations of mathematics. For example, Von Neumann–Bernays–Gödel set theory, is an extension of the set theory in which the collection of all sets is a class. This theory includes the replacement axiom, which may be stated as: If is a set and is a function, then is a set.


title: Function (mathematics)

text: In computer programming, a function is, in general, a piece of a computer program, which implements the abstract concept of function. That is, it is a program unit that produces an output for each input. However, in many programming languages every subroutine is called a function, even when there is no output, and when the functionality consists simply of modifying some data in the computer memory.


title: Function (mathematics)

text: Functional programming is the programming paradigm consisting of building programs by using only subroutines that behave like mathematical functions. For example, codice_1 is a function that takes three functions as arguments, and, depending on the result of the first function ("true" or "false"), returns the result of either the second or the third function. An important advantage of functional programming is that it makes easier program proofs, as being based on a well founded theory, the lambda calculus (see below).


title: Function (mathematics)

text: Except for computer-language terminology, "function" has the usual mathematical meaning in computer science. In this area, a property of major interest is the computability of a function. For giving a precise meaning to this concept, and to the related concept of algorithm, several models of computation have been introduced, the old ones being general recursive functions, lambda calculus and Turing machine. The fundamental theorem of computability theory is that these three models of computation define the same set of computable functions, and that all the other models of computation that have ever been proposed define the same set of computable functions or a smaller one. The Church–Turing thesis is the claim that every philosophically acceptable definition of a "computable function" defines also the same functions.


title: Function (mathematics)

text: Although defined only for functions from integers to integers, they can model any computable function as a consequence of the following properties:


title: Function (mathematics)

text: Lambda calculus is a theory that defines computable functions without using set theory, and is the theoretical background of functional programming. It consists of "terms" that are either variables, function definitions (""-terms), or applications of functions to terms. Terms are manipulated through some rules, (the -equivalence, the -reduction, and the -conversion), which are the axioms of the theory and may be interpreted as rules of computation.


title: Function (mathematics)

text: In its original form, lambda calculus does not include the concepts of domain and codomain of a function. Roughly speaking, they have been introduced in the theory under the name of "type" in typed lambda calculus. Most kinds of typed lambda calculi can define fewer functions than untyped lambda calculus.


title: Moby

text: Richard Melville Hall (born September 11, 1965), known professionally as Moby, is an American musician, songwriter, singer, producer, and animal rights activist. He has sold 20 million records worldwide. AllMusic considers him to be "among the most important dance music figures of the early 1990s, helping bring dance music to a mainstream audience both in the United States and the United Kingdom".


title: Moby

text: After taking up guitar and piano at age nine, he played in several underground punk rock bands through the 1980s before turning to electronic dance music. In 1989, he moved to New York City and became a prolific figure as a DJ, producer and remixer. His 1991 single "Go" was his mainstream breakthrough, especially in Europe, where it peaked within the top ten of the charts in the Netherlands and the United Kingdom. Between 1992 and 1997 he scored eight top 10 hits on the "Billboard" Dance Club Songs chart including "Move (You Make Me Feel So Good)", "Feeling So Real", and "James Bond Theme (Moby Re-Version)". Throughout the decade he also produced music under various pseudonyms, released the critically acclaimed "Everything Is Wrong" (1995), and composed music for films. His punk-oriented album "Animal Rights" (1996) alienated much of his fan base.


title: Moby

text: Moby found commercial and critical success with his fifth album "Play" (1999) which, after receiving little recognition, became an unexpected global hit in 2000 after each track was licensed to films, television shows, and commercials. It remains his highest selling album with 12 million copies sold. Its seventh single, "South Side", featuring Gwen Stefani, remains his only one to appear on the US "Billboard" Hot 100, reaching No. 14. Moby followed "Play" with albums of varied styles including electronic, dance, rock, and downtempo music, starting with "18" (2002), "Hotel" (2005), and "Last Night" (2008). His later albums saw him explore ambient music, including the almost four-hour release "" (2016). Moby continues to record and release albums; his nineteenth studio album, "Reprise", was released in May 2021.


title: Moby

text: In addition to his music career, Moby is known for his veganism and support for animal rights and humanitarian aid. He was the owner of TeaNY, a vegan cafe in Manhattan, and Little Pine, a vegan restaurant in Los Angeles, and organized the vegan music and food festival Circle V. He is the author of four books, including a collection of his photography and two memoirs: "" (2016) and "Then It Fell Apart" (2019).


title: Moby

text: Richard Melville Hall was born September 11, 1965, in the neighborhood of Harlem in Manhattan, New York City. He is an only child of Elizabeth McBride ("née" Warner), a medical secretary, and James Frederick Hall, a chemistry professor, who died in a car crash while drunk when Moby was two. His father gave him the nickname Moby three days after his birth as his parents considered the name Richard too large for a newborn baby. The name was also a reference to the family's ancestry; Hall says he is the great-great-great nephew of Herman Melville, author of "Moby-Dick".


title: Moby

text: Moby was raised by his mother, first in San Francisco from 1969 for a short period. He recalled being sexually abused by a staff member at his daycare during this time. This was followed by a move to Darien, Connecticut, living in a squat with "three or four other drug-addicted hippies, with bands playing in the basement." The two then moved to Stratford, Connecticut, for a brief time. His mother struggled to support her son, often relying on food stamps and government welfare. They occasionally stayed with Moby's grandparents in Darien, but the affluence of the New York City suburb made him feel poor and ashamed. Shortly before his mother's death in 1997, Moby learned from her that he has a half brother. His first job was a caddy at a golf course.


title: Moby

text: Moby took up music at the age of nine. He started on classical guitar and received piano lessons from his mother before studying jazz, music theory, and percussion. In 1983, he became the guitarist in a hardcore punk band, the Vatican Commandos, playing on their debut EP "Hit Squad for God". Around this time he was the lead vocalist for Flipper for two days; Moby played bass for their reunion shows in the 2000s. Moby formed a post punk group named AWOL around the time of his eighteenth birthday. He is credited on their only release, a self-titled EP, as Moby Hall.


title: Moby

text: In 1983, Moby graduated from Darien High School and started a philosophy degree at the University of Connecticut in Storrs, Connecticut. Around this time he had found the instruments he had learned "sonically limiting" and moved to electronic music. He spun records at the campus radio station WHUS, which led to DJ work in local clubs and bars. Moby grew increasingly unhappy at university, however, and transferred to State University of New York at Purchase, studying philosophy and photography, to try and renew his interest in studying. He dropped out in April 1984 to pursue DJing and music full-time, which started his interest in electronic dance music. For two years he lived in Greenwich, Connecticut, where he DJ'd at The Cafe, an under-21 nightclub at the back of a church. In 1987, he started to send demos of his music to record labels in New York City; he failed to receive an offer, which led to a two-year period of "very fruitless labor". Around 1988, Moby moved into a semi-abandoned factory in Stamford, Connecticut, that had no bathroom or running water, but the free electricity supply allowed him to work on his music, using a 4-track recorder, synthesizer, and drum machine.


title: Moby

text: Moby's formative musical influences include Nick Drake, Orchestral Manoeuvres in the Dark (OMD), Suicide, Silver Apples, Eric B. & Rakim, and Public Enemy.


title: Moby

text: In 1989, Moby relocated to New York City with his close friend, artist Damian Loeb. In addition to performing DJ sets in local bars and clubs, he played guitar in alternative rock group Ultra Vivid Scene and appeared in the video for their 1989 single "Mercy Seat". In 1990, Moby joined Shopwell and played on their album "Peanuts". Moby's first live electronic music gig followed in the summer of 1990 at Club MK; he wore a suit for the show. His future manager Eric Härle, who was in attendance, recalled Moby's set: "The music was amazing, but the show was riddled with technical mishaps. It left me very intrigued and impressed in a strange way."


title: Moby

text: By mid-1990, Moby had signed a deal as the sole artist of Instinct Records, an independent New York City-based dance label then still in its infancy. The three-man operation saw Moby answer incoming calls and make records in a studio he set up in the owner's lounge. To appear that Instinct had more artists, Moby's early singles were put out under several names such as Voodoo Child, Barracuda, Brainstorm, and UHF. The first, "Time's Up" as The Brotherhood, was co-written by Moby and vocalist Jimmy Mack. This was followed by "Mobility", his first single released as Moby, in November 1990 which sold an initial 2,000 copies. He then scored a breakthrough hit with a remix of "Go", originally a B-side to "Mobility" with an added sample of "Laura Palmer's Theme" by Angelo Badalamenti from the television series "Twin Peaks". Released in March 1991, it peaked at No. 10 in the UK in October and earned him national exposure there with an appearance on "Top of the Pops". Instinct capitalised on Moby's success with the late 1991 compilation "Instinct Dance" featuring tracks by Moby and his pseudonyms. The following year, Moby revealed that "Go" had earned him just $2,000 in royalties.


title: Moby

text: The success of "Go" led to increased demand for Moby to produce more music and to remix other artists' songs. He often arranged for the artist and himself to trade remixes as opposed to being paid for his work, which was the case for his mixes for Billy Corgan and Soundgarden. The increased mainstream exposure led Moby to request a release from his contract with Instinct for a bigger label. Instinct refused, so Moby retaliated by holding out on new material. However, Instinct continued to put out records, mostly from demos, without his consent having previously copied many of his tapes and had the master rights. This was the case for Moby's debut album, "Moby", released in July 1992 and formed mostly of previously unreleased demos that Moby considered old and unrepresentative of the musical direction he had taken since. Nonetheless, he claimed Instinct had insisted and had the legal right to put it out. It was re-titled "The Story So Far" and presented with a different track listing for its UK release. Four singles were released: "Go", "Drop a Beat", "Next Is the E", and a double A-side of "I Feel It" with "Thousand". The latter was recognised by "Guinness World Records" as the fastest tempo in a recorded song at 1,015 beats-per-minute.


title: Moby

text: In 1992, Moby completed his first US tour as the opening act for The Shamen. In mid-1992, Moby estimated that he had earned between $8,000 to $11,000 a year for the past six years. At the 1992 "Mixmag" awards, he smashed his keyboard after his set. After his second nationwide tour, this time with The Prodigy and Richie Hawtin, in early 1993, a second compilation of Moby's work for Instinct followed named "Early Underground". His second and final album on Instinct, "Ambient", was released in August 1993. It is a collection of mostly ambient techno instrumentals of a more experimental style. By this time Instinct had agreed to release Moby who then took legal action, claiming that the label demanded "a ridiculous amount of money" that he did not have to leave. He also expressed disagreements over the way Instinct had packaged and handled his music. Moby was eventually released after he paid the label $10,000.


title: Moby

text: In 1993, Moby signed with Elektra Records which lasted for five years. He secured a deal with Mute Records, a British label, to handle his European distribution. Moby's output for Elektra/Mute began with "Move", a four-track EP released in August 1993. He attempted to make it in a professional studio, but he disliked the results and re-recorded it at home. The song "All That I Need Is to Be Loved (MV)" is his first song to feature his own vocals. The first single, "Move (You Make Me Feel So Good)", reached No. 1 on the US "Billboard" Hot Dance Music/Club Play chart and No. 21 in the UK. In 1993, Moby toured as the headlining act with Orbital and Aphex Twin. A rift developed between Aphex Twin and himself, partly due to Moby's refusal to tolerate their cigarette smoke, so he travelled to each gig by plane, leaving the rest on the tour bus. In 1994, Moby put out "Demons/Horses", an electronic album of two 20-minute tracks under the name Voodoo Child.


title: Moby

text: Moby's contract with Elektra allowed the opportunity to make his third full-length album, which was underway in 1994. He chose to include a variety of musical styles on the album that he either liked or had been influenced by, including electronic dance, ambient, rock, and industrial music. "Everything Is Wrong" was released in March 1995 to critical praise; "Spin" magazine named it Album of the Year and some commentators considered it to be an album ahead of its time as it failed to crack the "Billboard" 200 or have an impact on the dance charts. In the UK, the album reached No. 25 and the singles "Hymn" and "Feeling So Real" went to Nos. 31 and 30, respectively. Elektra took advantage of its diverse sound by distributing tracks of the same style to corresponding radio stations nationwide. Early copies put out in the UK and Germany included a bonus CD of ambient music entitled "Underwater". Moby toured the album with some headline spots on the second stage at the 1995 Lollapalooza festival. He followed it with a double remix album, "Everything Is Wrong—Mixed and Remixed".