Inverse function

If f is invertible, then there is exactly one function g satisfying this property. The function g is called the inverse of f, and is usually denoted as f −1, a notation introduced by John Frederick William Herschel in 1813.[2][3][4][5][6][nb 1]

The inverse function f −1 to f can be explicitly described as the function

Recall that if f is an invertible function with domain X and codomain Y, then

Using the composition of functions, this statement can be rewritten to the following equations between functions:

where idX is the identity function on the set X; 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 f −1. Repeatedly composing a function f: XX with itself is called iteration. If f is applied n times, starting with the value x, then this is written as fn(x); so f 2(x) = f (f (x)), etc. Since f −1(f (x)) = x, composing f −1 and fn yields fn−1, "undoing" the effect of one application of f.

While the notation f −1(x) might be misunderstood,[1] (f(x))−1 certainly denotes the multiplicative inverse of f(x) and has nothing to do with the inverse function of f.[6]

The following table shows several standard functions and their inverses:

This allows one to easily determine inverses of many functions that are given by algebraic formulas. For example, if f is the function

Sometimes, the inverse of a function cannot be expressed by a closed-form formula. For example, if f is the function

then f is a bijection, and therefore possesses an inverse function f −1. The formula for this inverse has an expression as an infinite sum:

Since a function is a special type of binary relation, many of the properties of an inverse function correspond to properties of converse relations.

If an inverse function exists for a given function f, then it is unique.[12] This follows since the inverse function must be the converse relation, which is completely determined by f.

There is a symmetry between a function and its inverse. Specifically, if f is an invertible function with domain X and codomain Y, then its inverse f −1 has domain Y and image X, and the inverse of f −1 is the original function f. In symbols, for functions f:XY and f−1:YX,[12]

This statement is a consequence of the implication that for f to be invertible it must be bijective. The involutory nature of the inverse can be concisely expressed by[13]

Notice that the order of g and f have been reversed; to undo f followed by g, we must first undo g, and then undo f.

For example, let f(x) = 3x and let g(x) = x + 5. Then the composition g ∘ f 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,

More generally, a function f : XX is equal to its own inverse, if and only if the composition f ∘ f is equal to idX. Such a function is called an involution.

This is identical to the equation y = f(x) that defines the graph of f, except that the roles of x and y have been reversed. Thus the graph of f −1 can be obtained from the graph of f by switching the positions of the x and y axes. This is equivalent to reflecting the graph across the line y = x.[15][1]

The inverse function theorem states that a continuous function f 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 f′(x) = 3x2 + 1 is always positive.

If the function f is differentiable on an interval I and f′(x) ≠ 0 for each xI, then the inverse f −1 is differentiable on f(I).[16] If y = f(x), the derivative of the inverse is given by the inverse function theorem,

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 f : RnRn is invertible in a neighborhood of a point p as long as the Jacobian matrix of f at p is invertible. In this case, the Jacobian of f −1 at f(p) is the matrix inverse of the Jacobian of f at p.

Even if a function f is not one-to-one, it may be possible to define a partial inverse of f by restricting the domain. For example, the function

is not one-to-one, since x2 = (−x)2. However, the function becomes one-to-one if we restrict to the domain x ≥ 0, in which case

(If we instead restrict to the domain x ≤ 0, then the inverse is the negative of the square root of y.) 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 f, and the portions (such as x and −x) 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 y is called the principal value of f −1(y).

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 x (and more generally sin(x + 2πn) = sin(x) for every integer n). However, the sine is one-to-one on the interval [− π/2,  π/2], 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 −π/2 and π/2. The following table describes the principal branch of each inverse trigonometric function:[18]

Left and right inverses are not necessarily the same. If g is a left inverse for f, then g may or may not be a right inverse for f; and if g is a right inverse for f, then g is not necessarily a left inverse for f. For example, let f: R[0, ∞) denote the squaring map, such that f(x) = x2 for all x in R, and let g: [0, ∞)R denote the square root map, such that g(x) = x for all x ≥ 0. Then f(g(x)) = x for all x in [0, ∞); that is, g is a right inverse to f. However, g is not a left inverse to f, since, e.g., g(f(−1)) = 1 ≠ −1.

If f: XY, a left inverse for f (or retraction of f ) is a function g: YX such that composing f with g from the left gives the identity function[citation needed]:

Thus, g must equal the inverse of f on the image of f, but may take any values for elements of Y not in the image.

A function f is injective if and only if it has a left inverse or is the empty function.[citation needed]

A right inverse for f (or section of f ) is a function h: YX such that[citation needed]

A function f has a right inverse if and only if it is surjective (though constructing such an inverse in general requires the axiom of choice).

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.

If f: XY is any function (not necessarily invertible), the preimage (or inverse image) of an element yY is defined to be the set of all elements of X that map to y:

The preimage of y can be thought of as the image of y under the (multivalued) full inverse of the function f.

For example, take the function f: RR; xx2. This function is not invertible as it is not bijective, but preimages may be defined for subsets of the codomain, e.g.