Inverse function

A function and its inverse.

In mathematics, the inverse of a function ƒ is a function which "reverses" ƒ. That is, if the function ƒ maps x to y:

$f(x)=y\,$

then the inverse function ƒ^–1 maps y back to x:

$f^{-1}(y)=x{\text{.}}\,$

Not every function has an inverse—those that do are called invertible.

Note on functions

From the most general point of view, a function is a mapping that sends each element of a set X (called the domain) to a uniquely determined element of a set Y (sometimes called the codomain). The notation

ƒ: X → Y

means "ƒ is a function from the set X to the set Y."

Definition

If ƒ maps X to Y, then ƒ^–1 maps Y back to X.

If ƒ: X → Y, then the inverse of ƒ, written ƒ^–1: Y → X, is the function defined by the following rule:

$f^{-1}(y)=x\;\;\;\;{\text{if and only if}}\;\;\;\;f(x)=y{\text{.}}$

Example

If ƒ: {1, 2, 3, 4} → {7, 8, 9, 10} is the function defined by

ƒ(1) = 8,

ƒ(2) = 10,

ƒ(3) = 9,

ƒ(4) = 7,

then the inverse of ƒ is the function ƒ^–1: {7, 8, 9, 10} → {1, 2, 3, 4} defined by

ƒ–1(7) = 4,

ƒ–1(8) = 1,

ƒ–1(9) = 3,

ƒ–1(10) = 2.

Example

Let R be the set of all real numbers, and let ƒ: R → R be the function defined by the rule

$f(x)=x+8{\text{.}}\,$

Then the inverse of ƒ is defined by the rule

$f^{-1}(y)=y-8{\text{.}}\,$

There is a symmetry between a function and its inverse: if g is equal to the inverse of ƒ, then ƒ is equal to the inverse of g. That is, the inverse of the inverse of ƒ is the original function ƒ.

Only a function that is both one-to-one and onto (i.e. a bijective function) possesses an inverse, though any one-to-one function has an inverse defined on its range. A function that has an inverse is called invertible.

Example

The function ƒ: R → R defined by

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

is not invertible, since it is not one-to-one. For example ƒ(–3) = ƒ(3) = 9, so the value of ƒ^–1(9) is not uniquely determined. The square root function is a partial inverse to ƒ (see partial inverses below).

Inverses and composition

Characterization

If ƒ: X → Y is invertible, then

${\begin{array}{l}{\text{1. }}f^{-1}\left(\,f(x)\,\right)=x{\text{, for every }}x\in X{\text{, and}}\\[6pt]{\text{2. }}f\left(\,f^{-1}(y)\,\right)=y{\text{, for every }}y\in Y{\text{.}}\end{array}}$

These two statements are equivalent to the definition of inverses given above. They can be interpreted as statements about the compositions of ƒ and ƒ^–1. Specifically,

${\begin{array}{l}{\text{1. }}f^{-1}\circ f={\text{id}}_{X}{\text{, and}}\\[6pt]{\text{2. }}f\circ f^{-1}={\text{id}}_{Y}{\text{,}}\end{array}}$

where id_X and id_Y are the identity functions on the sets X and Y.

These identities are related to the origin of the notation ƒ^–1. If r is a nonzero real number, the multiplicative inverse of r is the reciprocal 1 / r = r ^–1. The multiplicative inverse satisfies the following identity:

$r^{-1}r=rr^{-1}=1\,$

The composition of two functions can be regarded as a kind of multiplication—indeed, many books use the notation ƒg for the composition of the functions ƒ and g. From this point of view, the inverse function is akin to a multiplicative inverse, which leads to the notation ƒ^–1 for the inverse of ƒ.

Inverse of a composition

The inverse of g o ƒ is ƒ^–1 o g^–1.

The inverse of a composition of functions is given by the formula

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

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

For example, let ƒ(x) = x + 5, and let g(x) = 3x. Then the composition ƒ o g is the function that first multiplies by three and then adds five:

$(f\circ g)(x)=3x+5$

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

$(f\circ g)^{-1}(y)={\frac {y-5}{3}}$

This is the composition (g^–1 o ƒ^–1) (y).

Inverses in calculus

Single-variable calculus is primarily concerned with functions that map real numbers to real numbers. Such functions are often defined through formulas, such as:

$f(x)=(2x+8)^{3}{\text{.}}\,$

A function ƒ from the real numbers to the real numbers possesses an inverse as long as it is one-to-one, i.e. as long as the graph of the function passes the horizontal line test. The domain of the inverse ƒ^–1 is the same as the range of the orginal function ƒ.

The following table shows several standard functions and their inverses:

Function ƒ(x)	Inverse ƒ–1(y)	Notes
x + a	y – a
a – x	a – y
mx	y / m	m ≠ 0
mx + b	(y – b) / m	m ≠ 0
1 / x	1 / y	x, y ≠ 0
x2	${\sqrt {y}}$	x, y ≥ 0 only, $\pm {\sqrt {y}}$ in general
x3	${\sqrt[{3}]{y}}$	no restriction on x and y
xp	x1/p (i.e. ${\sqrt[{p}]{x}}$)	x, y ≥ 0 in general, p ≠ 0
ax2 + bx + c	${\dfrac {-b\pm {\sqrt {b^{2}-4a(c-y)}}}{2a}}$	follows from the quadratic formula, two branches (corresponding to the plus and minus)
ex	ln y	y ≥ 0
ax	loga y	y ≥ 0 and a > 0
trigonometric functions	inverse trigonometric functions	various restrictions (see table below)

Formula for the inverse

A formula for ƒ^–1 can be found by solving the equation y = ƒ(x) for the variable x. For example, if ƒ is the function

$f(x)=(2x+8)^{3}{\text{.}}\,$

the we must solve the equation y = (2x + 8)³ for x:

${\begin{alignedat}{2}y&&&=(2x+8)^{3}\\[6pt]{\sqrt[{3}]{y}}&&&=2x+8\\[6pt]{\sqrt[{3}]{y}}-8&&&=2x\\[6pt]{\dfrac {{\sqrt[{3}]{y}}-8}{2}}&&&=x{\text{.}}\end{alignedat}}$

Thus the inverse function ƒ^–1 is given by the formula

$f^{-1}(y)={\dfrac {{\sqrt[{3}]{y}}-8}{2}}{\text{.}}\,$

Sometimes the inverse of a function cannot be expressed by a formula. For example, if ƒ is the function

$f(x)=x+\sin x{\text{,}}\,$

then ƒ is one-to-one, and therefore possesses an inverse function ƒ^–1. There is no simple formula for this inverse, since the equation y = x + sin x cannot be solved algebraically for x.

Graph of the inverse

The graphs of y = ƒ(x) and y = ƒ^–1(x). The dotted line is y = x.

If ƒ and ƒ^–1 are inverses, then the graph of the function

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

is the same as the graph of the equation

$x=f(y){\text{.}}\,$

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

Inverses and derivatives

A continuous function ƒ is one-to-one (and hence invertible) if and only if it is either increasing or decreasing (with no local maxima or minima). For example, the function

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

is invertible, since the derivative ƒ′(x) = 3x² + 1 is always positive.

If the function ƒ is differentiable, then the inverse ƒ^–1 will be differentiable as long as ƒ′(x) ≠ 0. The derivative of the inverse is given by the inverse function theorem:

${\frac {d}{dy}}\left[f^{-1}(y)\right]={\frac {1}{f'\left(f^{-1}(y)\right)}}{\text{.}}$

If we set x = ƒ^–1(y), then the formula above can be written

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

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 function ƒ: Rⁿ → Rⁿ is invertible in a neighborhood of a point p as long as the Jacobian matrix of ƒ at p is invertible. In this case, the Jacobian of ƒ^–1 at ƒ(p) is the matrix inverse of the Jacobian of ƒ at p.

Generalizations

Partial inverses

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

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

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

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

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

(If we instead restrict to the domain x ≤ 0, then the inverse is the negative of the square root of x.) Alternatively, there is no need to restrict the domain if we are content with the inverse being a multivalued function:

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

File:Inverse Cubic Graph.png

The inverse of this cubic function has three branches.

Sometimes this multivalued inverse is called the full inverse of ƒ, and the portions (such as ${\sqrt {x}}$ and $-{\sqrt {x}}$) are called branches. The most important branch of a multivalued function (e.g. the positive square root) is called the principle branch, and its value at y is called the principle value of ƒ^–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 picture to the right).

File:Arcsine Graph.png

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

$\sin(x+2\pi )=\sin(x)\,$

for every real x (and more generally sin(x + 2$\pi n$) = sin(x) for every integer $n$). However, the sine is one-to-one on the interval [–$\pi$/2, $\pi$/2], and the corresponding partial inverse is called the arcsine. This is considered the principle branch of the inverse sine, so the principle value of the inverse sine is always between –$\pi$/2 and $\pi$/2. The following table describes the principle branch of each inverse trigonometric function:

function	Range of usual principal value
sin–1	–$\pi /2$ ≤ sin–1(x) ≤ $\pi /2$
cos–1	0 ≤ cos–1(x) ≤ $\pi$
tan–1	–$\pi /2$ < tan–1(x) < $\pi /2$
cot–1	0 < cot–1(x) < $\pi$
sec–1	0 < sec–1(x) < $\pi$
csc–1	−$\pi /2$ ≤ csc–1(x) < $\pi /2$

Left and right inverses

If ƒ: X → Y, a left inverse for ƒ (or retraction of ƒ) is a function g: Y → X such that

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

That is, the function g satisfies the rule

${\text{If }}f(x)=y{\text{, then }}g(y)=x{\text{.}}\,$

Thus, g must equal the inverse of ƒ on the range of ƒ, but may take any values for elements of Y not in the range. A function ƒ has a left inverse if and only if it is one-to-one.

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

$f\circ h={\text{id}}_{Y}{\text{.}}\,$

That is, the function h satisfies the rule

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

Thus, h(y) may be any of the elements of x that map to y under ƒ. A function ƒ has a right inverse if and only if it is onto. (This statement is equivalent to the axiom of choice.)

Preimages

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

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

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 is the set of all elements of X that map to S:

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

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

See also

• Inverse trigonometric function
• Logarithm
• Inverse function theorem
• Inverse functions and differentiation
• Inverse relation
• Inverse element

References

• Stewart, James (2002), Calculus (5th ed.), Brooks Cole, ISBN 978-0534393397