In mathematical analysis, an inverse function is in simple terms a function which "does the reverse" of a given function. More formally, if f is a function with domain X, then f -1 is its inverse function if and only if for every we have:
f -1(f(x)) = f(f -1(x)) = x.
For example, if the function x → 3x + 2 is given, then its inverse function is x → (x - 2) / 3. This is usually written as:
- f : x → 3x + 2
- f -1 : x → (x - 2) / 3
The superscript "-1" is not an exponent. Similarly, f 2
) means "do f
twice", that is f
)), not the square of f
) (unfortunately, this notation has an exception for the trigonometric functions
) usually does
mean the square of sin(x
). As such, the prefix arc
is sometimes used to denote inverse trigonometric functions, eg arcsin x for the inverse of sin x).
Generally, if f(x) is any function, and g is its inverse, then g(f(x)) = x and f(g(x)) = x. In other words, an inverse function undoes what the original function does. In the above example, we can prove f -1 is the inverse by substituting (x - 2) / 3 into f, so
- 3(x - 2) / 3 + 2 = x.
Similarly this can be shown for substituting f
into f -1
For a function f to have a valid inverse, it must be a bijection, that is:
- each element in the codomain must be "hit" by f: otherwise there would be no way of defining the inverse of f for some elements
- each element in the codomain must be "hit" by f only once: otherwise the inverse function would have to send that element back to more than one value.
It is possible to work around this condition, by redefining f
's codomain to be precisely its range
, and by admitting a multi-valued function as an inverse.
If one represents the function f graphically in an x-y coordinate system, then the graph of f -1 is the reflection of the graph of f across the line y = x.
Algebraically, one computes the inverse function of f by solving the equation
- y = f(x)
, and then exchanging y
- y = f -1(x).
This is not always easy; if the function f
) is analytic
, the Lagrange inversion theorem
may be used.