Inverse function

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

${\displaystyle f^{-1}\left(f(x)\right)=x}$, for every ${\displaystyle x\in X}$ and ${\displaystyle f\left(f^{-1}(y)\right)=y}$ for every ${\displaystyle y\in Y}$.

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

${\displaystyle f^{-1}\circ f=\operatorname {id} _{X}}$ and ${\displaystyle f\circ f^{-1}=\operatorname {id} _{Y},}$

where id_X 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: X→X with itself is called iteration. If f is applied n times, starting with the value x, then this is written as fⁿ(x); so f²(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]

In keeping with the general notation, some English authors use expressions like sin⁻¹(x) to denote the inverse of the sine function applied to x (actually a partial inverse; see below).[7][6] Other authors feel that this may be confused with the notation for the multiplicative inverse of sin (x), which can be denoted as (sin (x))⁻¹.[6] To avoid any confusion, an inverse trigonometric function is often indicated by the prefix "arc" (for Latin arcus).[8][9] For instance, the inverse of the sine function is typically called the arcsine function, written as arcsin(x).[8][9] Similarly, the inverse of a hyperbolic function is indicated by the prefix "ar" (for Latin ārea).[9] For instance, the inverse of the hyperbolic sine function is typically written as arsinh(x). [9] Note that the expressions like sin⁻¹(x) can still be useful to distinguish the multivalued inverse from the partial inverse: ${\displaystyle \sin ^{-1}(x)=\{(-1)^{k}\arcsin(x)+\pi n:n\in \mathbb {Z} \}}$. Other inverse special functions are sometimes prefixed with the prefix "inv", if the ambiguity of the f⁻¹ notation should be avoided.[10][9]

The function f: R → [0,∞) given by f(x) = x² is not injective because ${\displaystyle (-x)^{2}=x^{2}}$ for all ${\displaystyle x\in \mathbb {R} }$. Therefore, f is not invertible.

If the domain of the function is restricted to the nonnegative reals, that is, we take the function ${\displaystyle f\colon [0,\infty )\to [0,\infty );\ x\mapsto x^{2}}$ with the same rule as before, then the function is bijective and so, invertible.[11] The inverse function here is called the (positive) square root function and is denoted by ${\displaystyle x\mapsto {\sqrt {x}}}$.

The following table shows several standard functions and their inverses:

Function f(x)	Inverse f −1(y)	Notes
x + a	y − a
a − x	a − y
mx	y/m	m ≠ 0
1/x (i.e. x−1)	1/y (i.e. y−1)	x, y ≠ 0
x2	${\displaystyle {\sqrt {y}}}$ (i.e. y1/2)	x, y ≥ 0 only
x3	${\displaystyle {\sqrt[{3}]{y}}}$ (i.e. y1/3)	no restriction on x and y
xp	${\displaystyle {\sqrt[{p}]{y}}}$ (i.e. y1/p)	x, y ≥ 0 if p is even; integer p > 0
2x	lb y	y > 0
ex	ln y	y > 0
10x	log y	y > 0
ax	loga y	y > 0 and a > 0
xex	W (y)	x ≥ −1 and y ≥ −1/e
trigonometric functions	inverse trigonometric functions	various restrictions (see table below)
hyperbolic functions	inverse hyperbolic functions	various restrictions

Many functions given by algebraic formulas possess a formula for their inverse. This is because the inverse ${\displaystyle f^{-1}}$ of an invertible function ${\displaystyle f\colon \mathbb {R} \to \mathbb {R} }$ has an explicit description as

${\displaystyle f^{-1}(y)=({\text{the unique element }}x\in \mathbb {R} {\text{ such that }}f(x)=y)}$.

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

${\displaystyle f(x)=(2x+8)^{3}}$

then to determine ${\displaystyle f^{-1}(y)}$ for a real number y, one must find the unique real number x such that (2x + 8)³ = y. This equation can be solved:

${\displaystyle {\begin{aligned}y&=(2x+8)^{3}\\{\sqrt[{3}]{y}}&=2x+8\\{\sqrt[{3}]{y}}-8&=2x\\{\dfrac {{\sqrt[{3}]{y}}-8}{2}}&=x.\end{aligned}}}$

Thus the inverse function f⁻¹ is given by the formula

${\displaystyle f^{-1}(y)={\frac {{\sqrt[{3}]{y}}-8}{2}}.}$

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

${\displaystyle f(x)=x-\sin x,}$

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:

${\displaystyle f^{-1}(y)=\sum _{n=1}^{\infty }{\frac {y^{n/3}}{n!}}\lim _{\theta \to 0}\left({\frac {\mathrm {d} ^{\,n-1}}{\mathrm {d} \theta ^{\,n-1}}}\left({\frac {\theta }{\sqrt[{3}]{\theta -\sin(\theta )}}}\right)^{n}\right).}$

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:X → Y and f⁻¹:Y → X,[12]

${\displaystyle f^{-1}\circ f=\operatorname {id} _{X}}$ and ${\displaystyle f\circ f^{-1}=\operatorname {id} _{Y}.}$

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]

${\displaystyle \left(f^{-1}\right)^{-1}=f.}$

The inverse of g ∘ f is f⁻¹ ∘ g⁻¹.

The inverse of a composition of functions is given by[14]

${\displaystyle (g\circ f)^{-1}=f^{-1}\circ g^{-1}.}$

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,

${\displaystyle (g\circ f)(x)=3x+5.}$

To reverse this process, we must first subtract five, and then divide by three,

${\displaystyle (g\circ f)^{-1}(x)={\tfrac {1}{3}}(x-5).}$

This is the composition (f⁻¹ ∘ g⁻¹)(x).

If X is a set, then the identity function on X is its own inverse:

${\displaystyle {\operatorname {id} _{X}}^{-1}=\operatorname {id} _{X}.}$

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

The graphs of y = f(x) and y = f⁻¹(x). The dotted line is y = x.

If f is invertible, then the graph of the function

${\displaystyle y=f^{-1}(x)}$

is the same as the graph of the equation

${\displaystyle x=f(y).}$

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

${\displaystyle f(x)=x^{3}+x}$

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 x ∈ I, 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,

${\displaystyle \left(f^{-1}\right)^{\prime }(y)={\frac {1}{f'\left(x\right)}}.}$

Using Leibniz's notation the formula above can be written as

${\displaystyle {\frac {dx}{dy}}={\frac {1}{dy/dx}}.}$

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 : Rⁿ → Rⁿ 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.

The square root of x is a partial inverse to f(x) = x².

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

${\displaystyle f(x)=x^{2}}$

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

${\displaystyle f^{-1}(y)={\sqrt {y}}.}$

(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:

${\displaystyle f^{-1}(y)=\pm {\sqrt {y}}.}$

The inverse of this cubic function has three branches.

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).

The arcsine is a partial inverse of the sine function.

These considerations are particularly important for defining the inverses of trigonometric functions. For example, the sine function is not one-to-one, since

${\displaystyle \sin(x+2\pi )=\sin(x)}$

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]

Function composition on the left and on the right need not coincide. In general, the conditions

1. "There exists g such that g(f(x))=x" and
2. "There exists g such that f(g(x))=x"

imply different properties of 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: X → Y, a left inverse for f (or retraction of f ) is a function g: Y → X such that composing f with g from the left gives the identity function[19]

$${\displaystyle g\circ f=\operatorname {id} _{X}{\text{.}}}$$

That is, the function g satisfies the rule

If f(x)=y, then g(y)=x.

The function 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 with nonempty domain is injective if and only if it has a left inverse.[20] An elementary proof runs as follows:

• If g is the left inverse of f, and f(x) = f(y), then g(f(x)) = g(f(y)) = x = y.

• If nonempty f: X → Y is injective, construct a left inverse g: Y → X as follows: for all y ∈ Y, if y is in the image of f, then there exists x ∈ X such that f(x) = y. Let g(y) = x; this definition is unique because f is injective. Otherwise, let g(y) be an arbitrary element of X.

  For all x ∈ X, f(x) is in the image of f. By construction, g(f(x)) = x, the condition for a left inverse.

In classical mathematics, every injective function f with a nonempty domain necessarily has a left inverse; however, this may fail in constructive mathematics. For instance, a left inverse of the inclusion {0,1} → R of the two-element set in the reals violates indecomposability by giving a retraction of the real line to the set {0,1}.[21]

Example of right inverse with non-injective, surjective function

A right inverse for f (or section of f ) is a function h: Y → X such that

${\displaystyle f\circ h=\operatorname {id} _{Y}.}$

That is, the function h satisfies the rule

If ${\displaystyle \displaystyle h(y)=x}$, then ${\displaystyle \displaystyle f(x)=y.}$

Thus, h(y) may be any of the elements of X that map to y under f.

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).

If h is the right inverse of f, then f is surjective. For all ${\displaystyle y\in Y}$, there is ${\displaystyle x=h(y)}$ such that ${\displaystyle f(x)=f(h(y))=y}$.

If f is surjective, f has a right inverse h, which can be constructed as follows: for all ${\displaystyle y\in Y}$, there is at least one ${\displaystyle x\in X}$ such that ${\displaystyle f(x)=y}$ (because f is surjective), so we choose one to be the value of h(y).

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 ${\displaystyle g}$ is a left inverse and ${\displaystyle h}$ a right inverse of ${\displaystyle f}$, for all ${\displaystyle y\in Y}$, ${\displaystyle g(y)=g(f(h(y))=h(y)}$.

A function has a two-sided inverse if and only if it is bijective.

A bijective function f is injective, so it has a left inverse (if f is the empty function, ${\displaystyle f\colon \varnothing \to \varnothing }$ is its own left inverse). f is surjective, so it has a right inverse. By the above, the left and right inverse are the same.

If f has a two-sided inverse g, then g is a left inverse and right inverse of f, so f is injective and surjective.

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

${\displaystyle f^{-1}(\{y\})=\left\{x\in X:f(x)=y\right\}.}$

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

Similarly, if S is any subset of Y, the preimage of S, denoted ${\displaystyle f^{-1}(S)}$, is the set of all elements of X that map to S:

${\displaystyle f^{-1}(S)=\left\{x\in X:f(x)\in S\right\}.}$

For example, take the function f: R → R; x ↦ x². This function is not invertible as it is not bijective, but preimages may be defined for subsets of the codomain, e.g.

${\displaystyle f^{-1}(\left\{1,4,9,16\right\})=\left\{-4,-3,-2,-1,1,2,3,4\right\}}$.

The preimage of a single element y ∈ Y – a singleton set {y}  – is sometimes called the fiber of y. When Y is the set of real numbers, it is common to refer to f⁻¹({y}) as a level set.