Inverse function
In mathematics, the inverse function of a function (also called the inverse of ) is a function that undoes the operation of . The inverse of exists if and only if is bijective, and if it exists, is denoted by formula_1 For a function formula_2, its inverse formula_3 admits an explicit description: it sends each element formula_4 to the unique element formula_5 such that . As an example, consider the real-valued function of a real variable given by . One can think of as the function which multiplies its input by 5 then subtracts 7 from the result. To undo this , one adds 7 to the input, then divides the result by 5. Therefore, the inverse of is the function formula_6 defined by formula_7 Definitions. Let be a function whose domain is the set , and whose codomain is the set . Then is "invertible" if there exists a function from to such that formula_8 for all formula_5 and formula_10 for all formula_4. If is invertible, then there is exactly one function satisfying this property. The function is called the inverse of , and is usually denoted as , a notation introduced by John Frederick William Herschel in 1813. The function is invertible if and  only if it is bijective. This is because the condition formula_8 for all formula_5 implies that is injective, and the condition formula_10 for all formula_4 implies that is surjective. The inverse function to can be explicitly described as the function Inverses and composition. Recall that if is an invertible function with domain and codomain , then Using the composition of functions, this statement can be rewritten to the following equations between functions: where is the identity function on the set ; that is, the function that leaves its argument unchanged. In category theory, this statement is used as the definition of an  inverse morphism. Considering function composition helps to understand the notation . Repeatedly composing a function with itself is called iteration. If is applied times, starting with the value , then this is written as ; so , etc. Since , composing and yields , "undoing" the effect of one application of . Notation. While the notation might be misunderstood, certainly denotes the multiplicative inverse of and has nothing to do with the inverse function of . In keeping with the general notation, some English authors use expressions like to denote the inverse of the sine function applied to (actually a partial inverse; see below). Other authors feel that  this may be confused with the notation for the multiplicative inverse of , which can be denoted as . To avoid any confusion, an inverse trigonometric function is often indicated by the prefix "arc" (for Latin ). For instance, the inverse of the sine function is typically called the arcsine function, written as . Similarly, the inverse of a hyperbolic function is indicated by the prefix "ar" (for Latin ). For instance, the inverse of the hyperbolic sine function is typically written as . Note that the expressions like can still be useful to distinguish the multivalued inverse from the partial inverse: formula_23. Other inverse special  functions are sometimes prefixed with the prefix "inv", if the ambiguity of the notation should be avoided. Examples. Squaring and square root functions. The function given by is not injective because formula_24 for all formula_25. Therefore, is not invertible. If the domain of the function is restricted to the nonnegative reals, that is, we take the function formula_26 with the same "rule" as before, then the function is bijective and so, invertible. The inverse function here is called the "(positive) square root function" and is denoted by formula_27. Standard inverse functions. The following table shows several standard functions and their  inverses: Formula for the inverse. Many functions given by algebraic formulas possess a formula for their inverse. This is because the inverse formula_28 of an invertible function formula_29 has an explicit description as This allows one to easily determine inverses of many functions that are given by algebraic formulas. For example, if is the function then to determine formula_32 for a real number , one must find the unique real number such that . This equation can be solved: Thus the inverse function is given by the formula Sometimes, the inverse of a function cannot be expressed by a closed-form  formula. For example, if is the function then is a bijection, and therefore possesses an inverse function . The formula for this inverse has an expression as an infinite sum: Properties. Since a function is a special type of binary relation, many of the properties of an inverse function correspond to properties of converse relations. Uniqueness. If an inverse function exists for a given function , then it is unique. This follows since the inverse function must be the converse relation, which is completely determined by . Symmetry. There is a symmetry between a function and its inverse. Specifically, if is an invertible  function with domain and codomain , then its inverse has domain and image , and the inverse of is the original function . In symbols, for functions and , This statement is a consequence of the implication that for to be invertible it must be bijective. The involutory nature of the inverse can be concisely expressed by The inverse of a composition of functions is given by Notice that the order of and have been reversed; to undo followed by , we must first undo , and then undo . For example, let and let . Then the composition is the function that first multiplies by three and  then adds five, To reverse this process, we must first subtract five, and then divide by three, This is the composition Self-inverses. If is a set, then the identity function on is its own inverse: More generally, a function is equal to its own inverse, if and only if the composition is equal to . Such a function is called an involution. Graph of the inverse. If is invertible, then the graph of the function is the same as the graph of the equation This is identical to the equation that defines the graph of , except that the roles of  and have been reversed. Thus the graph of can be obtained from the graph of by switching the positions of the and axes. This is equivalent to reflecting the graph across the line Inverses and derivatives. The inverse function theorem states that a continuous function is invertible on its range (image) if and only if it is either strictly increasing or decreasing (with no local maxima or minima). For example, the function is invertible, since the derivative If the function is differentiable on an interval and for each , then the inverse is differentiable on . If , the derivative of the inverse  is given by the inverse function theorem, Using Leibniz's notation the formula above can be written as This result follows from the chain rule (see the article on inverse functions and differentiation). The inverse function theorem can be generalized to functions of several variables. Specifically, a differentiable multivariable function is invertible in a neighborhood of a point as long as the Jacobian matrix of at is invertible. In this case, the Jacobian of at is the matrix inverse of the Jacobian of at . Generalizations. Partial inverses. Even if a function is not one-to-one, it may be possible  to define a partial inverse of by restricting the domain. For example, the function is not one-to-one, since . However, the function becomes one-to-one if we restrict to the domain , in which case (If we instead restrict to the domain , then the inverse is the negative of the square root of .) Alternatively, there is no need to restrict the domain if we are content with the inverse being a multivalued function: Sometimes, this multivalued inverse is called the full inverse of , and the portions (such as and −) are called "branches". The most important branch of a  multivalued function (e.g. the positive square root) is called the "principal branch", and its value at is called the "principal value" of . For a continuous function on the real line, one branch is required between each pair of local extrema. For example, the inverse of a cubic function with a local maximum and a local minimum has three branches (see the adjacent picture). These considerations are particularly important for defining the inverses of trigonometric functions. For example, the sine function is not one-to-one, since for every real (and more generally for every integer ). However, the sine is one -to-one on the interval , and the corresponding partial inverse is called the arcsine. This is considered the principal branch of the inverse sine, so the principal value of the inverse sine is always between − and . The following table describes the principal branch of each inverse trigonometric function: Left and right inverses. Left and right inverses are not necessarily the same. If is a left inverse for , then may or may not be a right inverse for ; and if is a right inverse for , then is not necessarily a left inverse for . For example, let denote the squaring map , such that for all in , and let denote the square root map, such that for all . Then for all in ; that is, is a right inverse to . However, is not a left inverse to , since, e.g., . Left inverses. If , a left inverse for (or "retraction" of ) is a function such that composing with from the left gives the identity function: formula_57 That is, the function satisfies the rule Thus, must equal the inverse of on the image of , but may take any values for elements of not in the image. A function is injective if and only if it has  a left inverse or is the empty function. In classical mathematics, every injective function with a nonempty domain necessarily has a left inverse; however, this may fail in constructive mathematics. For instance, a left inverse of the inclusion of the two-element set in the reals violates indecomposability by giving a retraction of the real line to the set . Right inverses. A right inverse for (or "section" of ) is a function such that That is, the function satisfies the rule Thus, may be any of the elements of that map to under . A function has a right inverse if and  only if it is surjective (though constructing such an inverse in general requires the axiom of choice). Two-sided inverses. An inverse that is both a left and right inverse (a two-sided inverse), if it exists, must be unique. In fact, if a function has a left inverse and a right inverse, they are both the same two-sided inverse, so it can be called the inverse. A function has a two-sided inverse if and only if it is bijective. Preimages. If is any function (not necessarily invertible), the preimage (or inverse image) of an element is defined  to be the set of all elements of that map to : The preimage of can be thought of as the image of under the (multivalued) full inverse of the function . Similarly, if is any subset of , the preimage of , denoted formula_77, is the set of all elements of that map to : For example, take the function . This function is not invertible as it is not bijective, but preimages may be defined for subsets of the codomain, e.g. The preimage of a single element – a singleton set – is sometimes called the "fiber" of . When is the set of real numbers, it  is common to refer to as a "level set".