Functions:Inverses

A function f and its inverse f⁻¹. Because f maps a to 3, the inverse f⁻¹ maps 3 back to a.

In mathematics, an inverse function (or anti-function) is a function that "reverses" another function: if the function f applied to an input x gives a result of y, then applying its inverse function g to y gives the result x, i.e., g(y) = x if and only if f(x) = y. The inverse function of f is also denoted as $f^{-1}$.

As an example, consider the real-valued function of a real variable given by f(x) = 5x − 7. Thinking of this as a step-by-step procedure (namely, take a number x, multiply it by 5, then subtract 7 from the result), to reverse this and get x back from some output value, say y, we would undo each step in reverse order. In this case, it means to add 7 to y, and then divide the result by 5. In functional notation, this inverse function would be given by,

$g(y)={\frac {y+7}{5}}.$

With y = 5x − 7 we have that f(x) = y and g(y) = x.

Not all functions have inverse functions. Those that do are called invertible. For a function f: X → Y to have an inverse, it must have the property that for every y in Y, there is exactly one x in X such that f(x) = y. This property ensures that a function g: Y → X exists with the necessary relationship with f.

Definitions

If f maps X to Y, then f⁻¹ maps Y back to X.

Let f be a function whose domain is the set X, and whose codomain is the set Y. Then f is invertible if there exists a function g with domain Y and codomain X, with the property:

$f(x)=y\,\,\Leftrightarrow \,\,g(y)=x.$

If f is invertible, then the function g is unique, which means that there is exactly one function g satisfying this property. Moreover, it also follows that the ranges of g and f equal their respective codomains. 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.

Stated otherwise, a function, considered as a binary relation, has an inverse if and only if the converse relation is a function on the codomain Y, in which case the converse relation is the inverse function.

Not all functions have an inverse. For a function to have an inverse, each element y ∈ Y must correspond to no more than one x ∈ X; a function f with this property is called one-to-one or an injection. If f⁻¹ is to be a function on Y, then each element y ∈ Y must correspond to some x ∈ X. Functions with this property are called surjections. This property is satisfied by definition if Y is the image of f, but may not hold in a more general context. To be invertible, a function must be both an injection and a surjection. Such functions are called bijections. The inverse of an injection f: X → Y that is not a bijection (that is, not a surjection), is only a partial function on Y, which means that for some y ∈ Y, f ⁻¹(y) is undefined. If a function f is invertible, then both it and its inverse function f⁻¹ are bijections.

Another convention is used in the definition of functions, referred to as the "set-theoretic" or "graph" definition using ordered pairs, which makes the codomain and image of the function the same. Under this convention, all functions are surjective, so bijectivity and injectivity are the same. Authors using this convention may use the phrasing that a function is invertible if and only if it is an injection. The two conventions need not cause confusion, as long as it is remembered that in this alternate convention, the codomain of a function is always taken to be the image of the function.

Example: Squaring and square root functions

The function f: R → [0,∞) given by f(x) = x² is not injective, since each possible result y (except 0) corresponds to two different starting points in X – one positive and one negative, and so this function is not invertible. With this type of function, it is impossible to deduce a (unique) input from its output. Such a function is called non-injective or, in some applications, information-losing.

If the domain of the function is restricted to the nonnegative reals, that is, the function is redefined to be f: [0, ∞) → [0, ∞) 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.

Inverses and composition

If f is an invertible function with domain X and codomain Y, then

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

Using the composition of functions, we can rewrite this statement as follows:

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

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 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.

Notation

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

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). 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. To avoid any confusion, an inverse trigonometric function is often indicated by the prefix "arc" (for Latin arcus). For instance, the inverse of the sine function is typically called the arcsine function, written as arcsin(x). Similarly, the inverse of a hyperbolic function is indicated by the prefix "ar" (for Latin ārea). For instance, the inverse of the hyperbolic sine function is typically written as arsinh(x). Other inverse special functions are sometimes prefixed with the prefix "inv", if the ambiguity of the f⁻¹ notation should be avoided.

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 f, then it is unique. This follows since the inverse function must be the converse relation, which is completely determined by f.

Symmetry

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,

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

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

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

The inverse of g ∘ f is f⁻¹ ∘ g⁻¹.

The inverse of a composition of functions is given by

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

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,

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

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

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

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

Self-inverses

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

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

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.

Generalizations

Partial inverses

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

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

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

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

(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:

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

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 ${\sqrt {x}}$ and −${\sqrt {x}}$) are called branches. The most important branch of a multivalued function (e.g. the positive square root) is called the 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

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

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 $[{\frac {-\pi }{2}}$, ${\frac {\pi }{2}}]$, 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 −${\frac {\pi }{2}}$ and ${\frac {\pi }{2}}$. The following table describes the principal branch of each inverse trigonometric function:

function	Range of usual principal value
arcsin	-${\frac {\pi }{2}}$ ≤ sin−1(x) ≤ ${\frac {\pi }{2}}$
arccos	0 ≤ cos−1(x) ≤ $\pi$
arctan	-${\frac {\pi }{2}}$ < tan−1(x) < ${\frac {\pi }{2}}$
arccot	0 < cot−1(x) < $\pi$
arcsec	0 ≤ sec−1(x) ≤ $\pi$
arccsc	-${\frac {\pi }{2}}$ ≤ csc−1(x) ≤ ${\frac {\pi }{2}}$

Left and right inverses

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 {{math|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) = ${\sqrt {x}}$ 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.

Left inverses

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:

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

That is, the function g satisfies the rule

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

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.

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

If f: X→Y is injective, f either is the empty function (X = ∅) or has a left inverse g: Y → X (X ≠ ∅), which can be constructed as follows: for all y ∈ Y, if y is in the image of f (there exists x ∈ X such that f(x) = y), let g(y) = x (x 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, so g(f(x)) = x by above, so g is a left inverse of f.

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} .

Right inverses

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

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

That is, the function h satisfies the rule

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

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 $y\in Y$, there is $x=h(y)$ such that $f(x)=f(h(y))=y$.

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

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.

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

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, $f\colon \emptyset \to \emptyset$ 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.

Preimages

If f: 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:

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

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 $f^{-1}(S)$, is the set of all elements of X that map to S:

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

For example, take a function f: R → R, where f: x ↦ x². This function is not invertible for reasons discussed in Example: Squaring and square root functions. Yet preimages may be defined for subsets of the codomain:

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

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.

Licensing

Content obtained and/or adapted from:

• Inverse function, Wikipedia under a CC BY-SA license