So a function is like a machine, that takes values of "x" and returns an output "y". The set of all values that "x" can have is called the "domain", and the set that contains every value that "y" can have is called the "codomain". A function is often denoted by italic letters such as formula_1, formula_2, formula_3.
If this happens, then we say that "y" is a function of "x, "and we write formula_4. Here, "formula_1" is the name of the function, and one writes formula_6 (function from X to Y) to represent the three parts of the function: the domain (x), the codomain (y), and the pairing process (the arrow).
An example of a function is formula_7. One gives a natural number formula_8 (0,1,2,3...) as the input, and gets a natural number formula_9, which is formula_8 +1 (1,2,3,4...) The idea of a function has been set up to cover all sorts of possibilities. The function doesn't have to be an equation. The main idea is that inputs and outputs are paired up somehow—even if the process might be very complicated.
The inputs and outputs can be put in a table like the picture; this is easy if there is not too much data.
In the picture, one can see that both 2 and 3 have been paired with c; this is not allowed in the other direction, as 2 could not output c and d at the same time (each input can only have one output).
All of the formula_11 (c and d in the picture) are usually called the "image" set of formula_1, and the image set can be the entire codomain or one of its subset. One can say that the image set of a subset A of the domain is f(A).
If the inputs and outputs have an order, then it is easy to plot them on a graph:
In that way, the image comes on the image of the set A.
In the 1690s, Gottfried Leibniz and Johann Bernoulli used the word "function" in letters between them, so the modern concept began at the same time as calculus.
In 1748, Leonhard Euler gave the following definition for function:"A function of a variable quantity is an analytic expression composed in any way whatsoever of the variable quantity and numbers or constant quantities."and then in 1755:"If some quantities so depend on other quantities that if the latter are changed the former undergoes change, then the former quantities are called functions of the latter. This definition applies rather widely and includes all ways in which one quantity could be determined by other. If, therefore, x denotes a variable quantity, then all quantities which depend upon x in any way, or are determined by it, are called functions of x."Usually, Peter Dirichlet is credited with the first modern definition of function (formulated in 1837). It is often used in schools until the second half of the 20th century:"y is a function of a variable x, defined on the interval a < x < b, if to every value of the variable x in this interval there corresponds a definite value of the variable y. Also, it is irrelevant in what way this correspondence is established."In 1939, the Bourbaki generalized the Dirichlet definition, and gave a set-theoretic version of the definition as a correspondence between inputs and outputs; this was used in schools from about 1960.
Finally in 1970, Bourbaki gave the modern definition as a triple formula_13, with formula_14 (i.e. formula_15 and formula_16).
Miami
Miami is a city located on the southeastern part of the U.S. state of Florida. It is a known tourist stop and it is well known for its Cuban, Puerto Rican and Haitian culture. Miami has one of the largest Hispanic communities in the United States, with over 70% of the population being of Hispanic and Latino American descent. Miami is also the county seat (and largest city) of Miami-Dade County.
Miami is nicknamed "The Magic City" because of its rapid growth.
The Tequesta tribe lived in what is now Miami before Spain claimed it. In 1566, Pedro Menéndez de Avilés, Florida's first governor, claimed the area from the Tequesta for Spain. Spain ruled Florida until Spain gave it to the United States in 1821. Florida became a state in 1845. On July 28, 1896, Miami officially became a city. It was named after the Mayaimi tribe.
Miami experienced rapid growth during the Florida land boom of the 1920s. However, this land boom was affected by the 1926 Miami hurricane. Many cubans migrated to Miami after Fidel Castro rose to power in Cuba following the 1959 Cuban Revolution, and the population rapidly increased. The majority of these Cuban immigrants now live in Little Havana.
During the 1980s, Miami was a primary location for drug trafficking from South America, mainly Colombia. This brought in millions of dollars into the city's economy, and it allowed the construction of luxury hotels, skyscrapers, nightclubs and car dealerships.
Miami has a tropical monsoon climate. Summers have average temperatures of during the day and at night, with average monthly rainfall around . Winters have average temperatures of during the day and at night, with average monthly rainfall around .
Miami-Dade County Public Schools is the school district for public schools in Miami. There are also some private schools there. There are many colleges and universities in and around Miami. A few of these include: Florida International University, University of Miami, Miami Dade College, and Florida Atlantic University.
There are several professional sports teams that play in and around Miami, including the Miami Marlins of Major League Baseball, the Miami Heat of the National Basketball Association, the Florida Panthers of the National Hockey League, the Miami Dolphins of the National Football League, and Inter Miami CF of Major League Soccer.
Also, three universities in and around Miami are members of NCAA Division I, the top division of college sports in the U.S.:
The population of Miami, Florida as of 2011 is 1.3 Million, with 41.4% suffering from poverty. According to the DNA North American Studies Institute, the racial-makeup of Miami is:
<nowiki>**</nowiki>Others** includes Asians, Arabs, and among others.
Grunge music
Grunge music or Seattle sound (as it is called sometimes) is a style of rock music that has elements of punk rock and heavy metal. It started in the mid- 1980s in the Pacific Northwest of the United States, especially in the states of Washington and Oregon. It became more popular in the early 1990s. Grunge songs often had lyrics with nihilism, depression, unhappiness, or apathy as themes. The widespread use of heroin as a recreational drug found its way into the lyrics and careers of numerous grunge rockers, notably Kurt Cobain and Layne Staley of "Alice in Chains".
The songs often had light (or clean) guitar sounds in the verses, and the high energy boost of distorted (or dirty) guitars in the choruses. The line-up for these bands was the typical rock band of guitars, bass, drums and vocal.
Although grunge as a movement became less popular by the last 1990s, its influence is still heard in many bands. Grunge musical sounds were copied by post-grunge bands such as Foo Fighters and Nickelback and nu metal bands such as Korn and Limp Bizkit.
Grunge music uses music from punk rock and heavy metal. Grunge guitar players use distorted electric guitar. Like punk and heavy metal, grunge bands use electric guitar, bass guitar, a drummer and a singer.
Some well known grunge artists were:
Carly Simon
Carly Simon (born June 25, 1945) is an American singer-songwriter and occasional children's book author. She was born in New York City to businessman Richard L. Simon (1899-1960) and Andrea Heinemann Simon (1909-1994). She was raised in Riverdale, The Bronx. She has two sisters: singer Joanna Simon (born 1940) and composer Lucy Simon (born 1943). She also has a brother, photographer Peter Simon.
Simon's songs include: "You're So Vain", "Nobody Does It Better", "Coming Around Again" and "Let the River Run".
Toto (band)
Toto is a soft rock group that formed in Los Angeles in 1977, that was popular in the early 1980's. Their best known song is "Africa" (1982).