# Inverse function

An inverse function is a function that reverses the mapping of another function: if \( f: X \to Y \) assigns to each \( x \in X \) a unique value \( f(x) \), its inverse \( f^{-1}: Y \to X \) sends each \( y \in Y \) back to the unique \( x \) with \( f(x) = y \). The defining property is that the two compositions recover the identity functions on the two sets: \( f^{-1} \circ f = \mathrm{id}_X \) and \( f \circ f^{-1} = \mathrm{id}_Y \).<sup>[1](https://encyclopediaofmath.org/wiki/Inverse_function)</sup> In words, the inverse undoes whatever the original function did.<sup>[2](https://www.csd.uwo.ca/~abrandt5/teaching/DiscreteStructures/Chapter3/functions.html)</sup>

| Key fact | Detail |
|---|---|
| Defining property | \( f^{-1}(f(x)) = x \) for all \( x \in X \) and \( f(f^{-1}(y)) = y \) for all \( y \in Y \)<sup>[1](https://encyclopediaofmath.org/wiki/Inverse_function)</sup> |
| Existence | A function has an inverse if and only if it is bijective, that is, both injective (one-to-one) and surjective (onto)<sup>[3](https://www.homepages.ucl.ac.uk/~ucahmto/0007/_book/1-3-functions.html)</sup> |
| Uniqueness | If an inverse exists, it is unique, which justifies the notation \( f^{-1} \)<sup>[4](https://personal.math.ubc.ca/~PLP/book/sec_inv_func.html)</sup> |
| Composition rule | If \( f \) and \( g \) are both invertible, then \( g \circ f \) is invertible and \( (g \circ f)^{-1} = f^{-1} \circ g^{-1} \)<sup>[3](https://www.homepages.ucl.ac.uk/~ucahmto/0007/_book/1-3-functions.html)</sup> |
| Graph facts | The domain of \( f \) equals the range of \( f^{-1} \) and vice versa; the two graphs are symmetric about the line \( y = x \)<sup>[5](https://math.umd.edu/~tjp/115%2002.8a%20handout%20fcn%20inverses.pdf)</sup> |
| Graphical test | A function has an inverse only if every horizontal line meets its graph at most once (the horizontal line test)<sup>[5](https://math.umd.edu/~tjp/115%2002.8a%20handout%20fcn%20inverses.pdf)</sup> |
| Derivative formula | If \( f \) is invertible and differentiable with nonzero derivative, \( (f^{-1})'(y) = 1/f'(f^{-1}(y)) \)<sup>[1](https://encyclopediaofmath.org/wiki/Inverse_function)</sup> |
| Notation origin | According to Cajori, the exponent \( -1 \) notation for inverse trigonometric functions was introduced by John Frederick William Herschel in 1813 in the *Philosophical Transactions* of London<sup>[6](https://math.stackexchange.com/questions/3396248/where-did-the-inverse-function-notation-f-1-come-from)</sup> |

## Existence and uniqueness

Not every function can be reversed. A function \( f: X \to Y \) is called injective, or one-to-one, if \( f(a) = f(b) \) implies \( a = b \); it is called surjective, or onto, if every element of \( Y \) is a value of \( f \). A function that is both is a bijection, and the central theorem of the subject states that \( f \) has an inverse function if and only if it is bijective.<sup>[3](https://www.homepages.ucl.ac.uk/~ucahmto/0007/_book/1-3-functions.html)</sup> The reason is visible in the definition: reversing a mapping requires each image to have a unique preimage, which is exactly injectivity, and every target value to be reached, which is surjectivity.<sup>[2](https://www.csd.uwo.ca/~abrandt5/teaching/DiscreteStructures/Chapter3/functions.html)</sup>

**Left and right inverses.** The two halves of bijectivity correspond to two one-sided notions. A left inverse of \( f \) is a function \( g: Y \to X \) with \( g \circ f = \mathrm{id}_X \); a right inverse is a function \( h: Y \to X \) with \( f \circ h = \mathrm{id}_Y \). A function has a left inverse if and only if it is injective, and a right inverse if and only if it is surjective.<sup>[3](https://www.homepages.ucl.ac.uk/~ucahmto/0007/_book/1-3-functions.html)</sup> When both exist they must coincide: for any \( y \in Y \), \( g(y) = g(f(h(y))) = h(y) \), so the two-sided inverse, when it exists, is unique.<sup>[3](https://www.homepages.ucl.ac.uk/~ucahmto/0007/_book/1-3-functions.html)</sup> The same uniqueness argument runs by composing one candidate inverse with the identity and then with the other, giving \( g = g \circ (f \circ h) = (g \circ f) \circ h = h \).<sup>[4](https://personal.math.ubc.ca/~PLP/book/sec_inv_func.html)</sup>

## Composition and algebraic structure

Composition is the operation against which inversion acts. If \( f: X \to Y \) and \( g: Y \to Z \), the composition \( g \circ f: X \to Z \) is defined by \( (g \circ f)(x) = g(f(x)) \), read as "do \( f \), then do \( g \)"; the order matters, and composition only makes sense when the codomain of \( f \) matches the domain of \( g \).<sup>[3](https://www.homepages.ucl.ac.uk/~ucahmto/0007/_book/1-3-functions.html)</sup>

**Reversal rule.** If \( f \) and \( g \) are both invertible, so is \( g \circ f \), and its inverse is \( f^{-1} \circ g^{-1} \), the original order reversed. The proof is a short associativity computation: \( (f^{-1} \circ g^{-1}) \circ (g \circ f) = f^{-1} \circ \mathrm{id}_Y \circ f = \mathrm{id}_X \), and similarly in the other order.<sup>[3](https://www.homepages.ucl.ac.uk/~ucahmto/0007/_book/1-3-functions.html)</sup>

The invertible maps from a set \( X \) to itself form a group under composition, with the identity map \( x \mapsto x \) as the neutral element and the inverse function as each element's group inverse.<sup>[1](https://encyclopediaofmath.org/wiki/Inverse_function)</sup> A map that is its own inverse, satisfying \( f \circ f = \mathrm{id} \), is called an involution; complex conjugation \( z = x + iy \mapsto x - iy \) on the complex plane is a standard example.<sup>[1](https://encyclopediaofmath.org/wiki/Inverse_function)</sup>

## Computing and graphing inverses

When a formula is available, the inverse is found by solving \( y = f(x) \) for \( x \) in terms of \( y \). For \( f(x) = 7x - 3 \) on the real line, this gives \( f^{-1}(y) = (y+3)/7 \), and the check \( f^{-1}(f(x)) = x \) confirms it.<sup>[4](https://personal.math.ubc.ca/~PLP/book/sec_inv_func.html)</sup> For a [Möbius transformation](https://www.edgepedia.org/mobius-transformation) \( h(x) = (ax-b)/(cx-d) \), the inverse is \( h^{-1}(y) = (dy-b)/(cy-a) \), with the two excluded points \( d/c \) and \( a/c \) swapping roles as the excluded values of domain and codomain.<sup>[4](https://personal.math.ubc.ca/~PLP/book/sec_inv_func.html)</sup>

**Graphical criteria.** On a graph, the test for invertibility is the horizontal line test: if any horizontal line intersects the graph more than once, the function is not one-to-one and has no inverse.<sup>[5](https://math.umd.edu/~tjp/115%2002.8a%20handout%20fcn%20inverses.pdf)</sup> When an inverse exists, the point \( (a, b) \) on the graph of \( f \) corresponds to \( (b, a) \) on the graph of \( f^{-1} \), so the two graphs are mirror images across the line \( y = x \), and the domain of one equals the range of the other.<sup>[5](https://math.umd.edu/~tjp/115%2002.8a%20handout%20fcn%20inverses.pdf)</sup>

**Domain restriction.** A function that fails the horizontal line test can often be made invertible by cutting its domain. The squaring function \( f(x) = x^2 \) has domain all real numbers and range \( [0, \infty) \), so it is not one-to-one; restricting the domain to \( [0, \infty) \) yields the inverse \( f^{-1}(y) = \sqrt{y} \), while restricting to \( (-\infty, 0] \) yields \( f^{-1}(y) = -\sqrt{y} \). The two choices produce two different inverse functions, and in each case the restricted domain of \( f \) equals the range of the chosen inverse.<sup>[7](https://math.utoledo.edu/~janders/1850/PreClassProblems/Problems26.pdf)</sup> This is the standard route to the inverse trigonometric functions, where each branch restriction of sine or cosine gives its own inverse on a chosen interval.

## Inverses in calculus

For real functions of one real variable, monotonicity replaces the abstract bijection condition. A continuous surjective function between intervals is injective if and only if it is strictly monotone, and in that case its inverse is also continuous.<sup>[1](https://encyclopediaofmath.org/wiki/Inverse_function)</sup> If \( f \) is differentiable with \( f'(y) \neq 0 \) everywhere, it is necessarily strictly monotone, and its inverse is differentiable as well.<sup>[1](https://encyclopediaofmath.org/wiki/Inverse_function)</sup>

**The derivative formula.** Differentiating the identity \( x = f(f^{-1}(x)) \) by the chain rule gives \( 1 = f'(f^{-1}(x)) \cdot (f^{-1})'(x) \), so

\[ (f^{-1})'(x) = \frac{1}{f'(f^{-1}(x))} \]

wherever \( f'(f^{-1}(x)) \neq 0 \).<sup>[8](https://math.libretexts.org/Courses/College_of_Southern_Nevada/Calculus_%28Hutchinson%29/03%3A_Differentiation/3.08%3A_Derivatives_of_Inverse_and_Logarithmic_Functions)</sup> Geometrically, the tangent lines to the two graphs at reflected points have reciprocal slopes, since reflection across \( y = x \) turns a slope of \( p/q \) into \( q/p \).<sup>[8](https://math.libretexts.org/Courses/College_of_Southern_Nevada/Calculus_%28Hutchinson%29/03%3A_Differentiation/3.08%3A_Derivatives_of_Inverse_and_Logarithmic_Functions)</sup> The formula produces familiar derivatives: applied to \( f(x) = x^3 \) it gives the derivative of the cube root, \( \frac{d}{dx}\sqrt[3]{x} = 1/(3x^{2/3}) \), and applied to \( f(x) = e^x \) it gives \( \frac{d}{dx}\ln x = 1/x \).<sup>[8](https://math.libretexts.org/Courses/College_of_Southern_Nevada/Calculus_%28Hutchinson%29/03%3A_Differentiation/3.08%3A_Derivatives_of_Inverse_and_Logarithmic_Functions)</sup>

**Several variables.** The one-variable picture generalizes through the inverse function theorem: if \( f \) is a continuously differentiable map on an open subset of \( \mathbb{R}^n \) whose differential (Jacobi matrix) is invertible at a point, then \( f \) is a local diffeomorphism near that point, that is, it has a continuously differentiable local inverse.<sup>[1](https://encyclopediaofmath.org/wiki/Inverse_function)</sup> The differential of the inverse is the inverse matrix of the differential, \( d(f^{-1})|_y = [df|_{f^{-1}(y)}]^{-1} \).<sup>[1](https://encyclopediaofmath.org/wiki/Inverse_function)</sup> Unlike the one-dimensional case, an everywhere-invertible differential does not guarantee global invertibility: the complex exponential \( (x, y) \mapsto (e^x \cos y, e^x \sin y) \) has invertible differential at every point but is not injective, since each nonzero target value has infinitely many preimages.<sup>[1](https://encyclopediaofmath.org/wiki/Inverse_function)</sup>

## Notation and history

The function symbol \( f(x) \) itself was first used by [Leonhard Euler](https://www.edgepedia.org/leonhard-euler) in 1734 in the *Commentarii Academiae Scientiarum Petropolitanae*.<sup>[9](https://math.hawaii.edu/~tom/history/functions.html)</sup> The inverse notation \( f^{-1} \) came much later. According to the historian Florian Cajori, the exponent \( -1 \) notation for inverse trigonometric functions was introduced by the astronomer John Frederick William Herschel in 1813 in the *Philosophical Transactions* of London, with a full-page footnote explaining his choice of forms such as \( \cos.^{-1} e \).<sup>[6](https://math.stackexchange.com/questions/3396248/where-did-the-inverse-function-notation-f-1-come-from)</sup> A calculus text by Daniel A. Murray (1908) adds that the notation had been explained earlier in Germany by an analyst named Burmann, citing Herschel's own 1820 collection of examples as a reference.<sup>[6](https://math.stackexchange.com/questions/3396248/where-did-the-inverse-function-notation-f-1-come-from)</sup>

The notation carries a known ambiguity. The same symbol \( f^{-1} \) is also widely used for the preimage, the set \( f^{-1}(\{y\}) = \{ x : f(x) = y \} \), which is defined for every function whether or not an inverse function exists; some authors even write \( f^{-1}(y) \) for this set.<sup>[1](https://encyclopediaofmath.org/wiki/Inverse_function)</sup> Context, and whether the argument is a point or a set, distinguishes the two uses.

## References

1. [Inverse function, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Inverse_function)
2. [3.1. Functions, Discrete Structures for Computing, Western University](https://www.csd.uwo.ca/~abrandt5/teaching/DiscreteStructures/Chapter3/functions.html)
3. [1.3 Functions, MATH0007: Algebra for Joint Honours Students, UCL](https://www.homepages.ucl.ac.uk/~ucahmto/0007/_book/1-3-functions.html)
4. [Inverse functions, Prelude to Mathematics, University of British Columbia](https://personal.math.ubc.ca/~PLP/book/sec_inv_func.html)
5. [Math 115 Chapter 2.8 Examples and Extra Notes, University of Maryland](https://math.umd.edu/~tjp/115%2002.8a%20handout%20fcn%20inverses.pdf)
6. [Where did the inverse function notation f⁻¹ come from?, Mathematics Stack Exchange (answer by Robert Israel)](https://math.stackexchange.com/questions/3396248/where-did-the-inverse-function-notation-f-1-come-from)
7. [Lesson X: Inverse functions and domain restriction, James D. Anderson, University of Toledo](https://math.utoledo.edu/~janders/1850/PreClassProblems/Problems26.pdf)
8. [3.8: Derivatives of Inverse and Logarithmic Functions, Mathematics LibreTexts](https://math.libretexts.org/Courses/College_of_Southern_Nevada/Calculus_%28Hutchinson%29/03%3A_Differentiation/3.08%3A_Derivatives_of_Inverse_and_Logarithmic_Functions)
9. [Earliest Uses of Function Symbols, University of Hawaii](https://math.hawaii.edu/~tom/history/functions.html)

---
*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Logic and discrete mathematics › Formal logic and foundations › Set theory › Elementary set theory*

*Initially written Sep 17, 2026 · Reviewed: Sep 30, 2026; Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*

*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*

License: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license