In mathematics, the inverse function of a function f (also called the inverse of f) is a function that undoes the operation of f. The inverse of f exists if and only if f is bijective, and if it exists, is denoted by
For a function , its inverse admits an explicit description: it sends each element to the unique element such that f(x) = y.
As an example, consider the real-valued function of a real variable given by f(x) = 5x − 7. One can think of f as the function which multiplies its input by 5 then subtracts 7 from the result. To undo this, one adds 7 to the input, then divides the result by 5. Therefore, the inverse of f is the function defined by
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 from Y to X such that for all and for all .^{[1]}
If f is invertible, then there is exactly one function g satisfying this property. 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.^{[2]}^{[3]}^{[4]}^{[5]}^{[6]}^{[nb 1]}
The function f is invertible if and only if it is bijective. This is because the condition for all implies that f is injective, and the condition for all implies that f is surjective.
The inverse function f^{ −1} to f can be explicitly described as the function
Recall that if f is an invertible function with domain X and codomain Y, then
Using the composition of functions, this statement can be rewritten to the following equations between functions:
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 f^{ n}(x); so f^{ 2}(x) = f (f (x)), etc. Since f^{ −1}(f (x)) = x, composing f^{ −1} and f^{ n} yields f^{ n−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]} The notation might be used for the inverse function to avoid ambiguity with the multiplicative inverse.^{[7]}
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).^{[8]}^{[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).^{[9]}^{[10]} For instance, the inverse of the sine function is typically called the arcsine function, written as arcsin(x).^{[9]}^{[10]} Similarly, the inverse of a hyperbolic function is indicated by the prefix "ar" (for Latin ārea).^{[10]} For instance, the inverse of the hyperbolic sine function is typically written as arsinh(x). ^{[10]} The expressions like sin^{−1}(x) can still be useful to distinguish the multivalued inverse from the partial inverse: . Other inverse special functions are sometimes prefixed with the prefix "inv", if the ambiguity of the f^{ −1} notation should be avoided.^{[11]}^{[10]}
The function f: R → [0,∞) given by f(x) = x^{2} is not injective because for all . Therefore, f is not invertible.
If the domain of the function is restricted to the nonnegative reals, that is, we take the function with the same rule as before, then the function is bijective and so, invertible.^{[12]} The inverse function here is called the (positive) square root function and is denoted by .
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 |
x^{2} | (i.e. y^{1/2}) | x, y ≥ 0 only |
x^{3} | (i.e. y^{1/3}) | no restriction on x and y |
x^{p} | (i.e. y^{1/p}) | x, y ≥ 0 if p is even; integer p > 0 |
2^{x} | lb y | y > 0 |
e^{x} | ln y | y > 0 |
10^{x} | log y | y > 0 |
a^{x} | log_{a} y | y > 0 and a > 0 |
xe^{x} | 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 of an invertible function has an explicit description as
This allows one to easily determine inverses of many functions that are given by algebraic formulas. For example, if f is the function
then to determine for a real number y, one must find the unique real number x such that (2x + 8)^{3} = y. This equation can be solved:
Thus the inverse function f^{ −1} is given by the formula
Sometimes, the inverse of a function cannot be expressed by a closed-form formula. For example, if f is the function
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:
Since a function is a special type of binary relation, many of the properties of an inverse function correspond to properties of converse relations.
If an inverse function exists for a given function f, then it is unique.^{[13]} 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,^{[13]}
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^{[14]}
The inverse of a composition of functions is given by^{[15]}
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,
To reverse this process, we must first subtract five, and then divide by three,
This is the composition (f^{ −1} ∘ g^{ −1})(x).
If X is a set, then the identity function on X is its own inverse:
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.
If f is invertible, then the graph of the function
is the same as the graph of the equation
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.^{[16]}^{[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
is invertible, since the derivative f′(x) = 3x^{2} + 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).^{[17]} If y = f(x), the derivative of the inverse is given by the inverse function theorem,
Using Leibniz's notation the formula above can be written as
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 : R^{n} → R^{n} 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.
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
is not one-to-one, since x^{2} = (−x)^{2}. However, the function becomes one-to-one if we restrict to the domain x ≥ 0, in which case
(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:
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).
These considerations are particularly important for defining the inverses of trigonometric functions. For example, the sine function is not one-to-one, since
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:^{[19]}
function | Range of usual principal value |
---|---|
arcsin | −π/2 ≤ sin^{−1}(x) ≤ π/2 |
arccos | 0 ≤ cos^{−1}(x) ≤ π |
arctan | −π/2 < tan^{−1}(x) < π/2 |
arccot | 0 < cot^{−1}(x) < π |
arcsec | 0 ≤ sec^{−1}(x) ≤ π |
arccsc | −π/2 ≤ csc^{−1}(x) ≤ π/2 |
Function composition on the left and on the right need not coincide. In general, the conditions
imply different properties of f. For example, let f: R → [0, ∞) denote the squaring map, such that f(x) = x^{2} 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^{[20]}
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.^{[21]} An elementary proof runs as follows:
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}.^{[22]}
A right inverse for f (or section of f ) is a function h: Y → X such that
That is, the function h satisfies the rule
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).
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.
A function has a two-sided inverse if and only if it is bijective.
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:
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 , is the set of all elements of X that map to S:
For example, take the function f: R → R; x ↦ x^{2}. This function is not invertible as it is not bijective, but preimages may be defined for subsets of the codomain, e.g.
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.
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[...] §473. Iterated logarithms [...] We note here the symbolism used by Pringsheim and Molk in their joint Encyclopédie article: "^{2}log_{b} a = log_{b} (log_{b} a), ..., ^{k+1}log_{b} a = log_{b} (^{k}log_{b} 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 d^{2} x, Δ^{3} x, Σ^{2} 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.=c^{x}. Some years later Herschel explained that in 1813 he used f^{n}(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 x^{2}, sin^{2} x. The prevailing notation at present is sin^{2} x, though the first is least likely to be misinterpreted. In the case of sin^{2} 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 log^{2} x, where log x · log x and log (log x) are of frequent occurrence in analysis. [...] The notation sin^{n} 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.)
α = arcsin m This notation is universally used in Europe and is fast gaining ground in this country. A less desirable symbol, α = sin^{-1} m, is still found in English and American texts. The notation α = inv sin m is perhaps better still on account of its general applicability. [...] A similar symbolic relation holds for the other trigonometric functions. It is frequently read 'arc-sine m' or 'anti-sine m', since two mutually inverse functions are said each to be the anti-function of the other.