Inverse Functions

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 {{math|f 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.

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.

Resources

• Inverse Functions, Lumen Learning
• Introduction to Inverse Function Video, Khan Academy

References

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9. Herschel, John Frederick William (1820). "Part III. Section I. Examples of the Direct Method of Differences". A Collection of Examples of the Applications of the Calculus of Finite Differences. Cambridge, UK: Printed by J. Smith, sold by J. Deighton & sons. pp. 1–13 [5–6]. Archived from the original on 2020-08-04. Retrieved 2020-08-04. [1] (NB. Inhere, Herschel refers to his 1813 work and mentions Hans Heinrich Bürmann's older work.)
10. Peirce, Benjamin (1852). Curves, Functions and Forces. I (new ed.). Boston, USA. p. 203.
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12. Cajori, Florian (1952) [March 1929]. "§472. The power of a logarithm / §473. Iterated logarithms / §533. John Herschel's notation for inverse functions / §535. Persistence of rival notations for inverse functions / §537. Powers of trigonometric functions". A History of Mathematical Notations. 2 (3rd corrected printing of 1929 issue, 2nd ed.). Chicago, USA: Open court publishing company. pp. 108, 176–179, 336, 346. ISBN 978-1-60206-714-1. Retrieved 2016-01-18. […] §473. Iterated logarithms […] We note here the symbolism used by Pringsheim and Molk in their joint Encyclopédie article: "2logb a = logb (logb a), …, k+1logb a = logb (klogb a)." […] §533. John Herschel's notation for inverse functions, sin−1 x, tan−1 x, etc., was published by him in the Philosophical Transactions of London, for the year 1813. He says (p. 10): "This notation cos.−1 e must not be understood to signify 1/cos. e, but what is usually written thus, arc (cos.=e)." He admits that some authors use cos.m A for (cos. A)m, but he justifies his own notation by pointing out that since d2 x, Δ3 x, Σ2 x mean dd x, ΔΔΔ x, ΣΣ x, we ought to write sin.2 x for sin. sin. x, log.3 x for log. log. log. x. Just as we write d−n V=∫n V, we may write similarly sin.−1 x=arc (sin.=x), log.−1 x.=cx. Some years later Herschel explained that in 1813 he used fn(x), f−n(x), sin.−1 x, etc., "as he then supposed for the first time. The work of a German Analyst, Burmann, has, however, within these few months come to his knowledge, in which the same is explained at a considerably earlier date. He[Burmann], however, does not seem to have noticed the convenience of applying this idea to the inverse functions tan−1, etc., nor does he appear at all aware of the inverse calculus of functions to which it gives rise." Herschel adds, "The symmetry of this notation and above all the new and most extensive views it opens of the nature of analytical operations seem to authorize its universal adoption."[a] […] §535. Persistence of rival notations for inverse function.— […] The use of Herschel's notation underwent a slight change in Benjamin Peirce's books, to remove the chief objection to them; Peirce wrote: "cos[−1] x," "log[−1] x."[b] […] §537. Powers of trigonometric functions.—Three principal notations have been used to denote, say, the square of sin x, namely, (sin x)2, sin x2, sin2 x. The prevailing notation at present is sin2 x, though the first is least likely to be misinterpreted. In the case of sin2 x two interpretations suggest themselves; first, sin x · sin x; second,[c] sin (sin x). As functions of the last type do not ordinarily present themselves, the danger of misinterpretation is very much less than in case of log2 x, where log x · log x and log (log x) are of frequent occurrence in analysis. […] The notation sinn x for (sin x)n has been widely used and is now the prevailing one. […] (xviii+367+1 pages including 1 addenda page) (NB. ISBN and link for reprint of 2nd edition by Cosimo, Inc., New York, USA, 2013.)
13. Smith, Eggen & St. Andre 2006, p. 202, Theorem 4.9
14. Wolf 1998, p. 198
15. Fletcher & Patty 1988, p. 116, Theorem 5.1
16. Wolf 1998, p. 208, Theorem 7.2
17. Smith, Eggen & St. Andre 2006, pg. 141 Theorem 3.3(a)
18. Lay 2006, p. 71, Theorem 7.26