Inverse function

An inverse function is a concept of mathematics. A function will calculate some output [math]\displaystyle{ y }[/math], given some input [math]\displaystyle{ x }[/math]. This is usually written [math]\displaystyle{ f(x) = y }[/math]. The inverse function does the reverse. Let's say [math]\displaystyle{ g }[/math] is the inverse function of [math]\displaystyle{ f }[/math], then [math]\displaystyle{ g(y) = x }[/math]. Or otherwise put, [math]\displaystyle{ g(f(x)) = x }[/math]. An inverse function to [math]\displaystyle{ f }[/math] is usually called [math]\displaystyle{ f^{-1} }[/math]. It is not to be confused with [math]\displaystyle{ 1/f }[/math], which is a reciprocal function.

Examples

If [math]\displaystyle{ f(x) = x^3 }[/math] over real [math]\displaystyle{ x }[/math], then [math]\displaystyle{ f^{-1}(x) = \sqrt[3]{x}. }[/math]

To find the inverse function, swap the roles of [math]\displaystyle{ x }[/math] and [math]\displaystyle{ y }[/math] and solve for [math]\displaystyle{ y }[/math]. For example, [math]\displaystyle{ y=e^x }[/math] would turn to [math]\displaystyle{ x=e^y }[/math], and then [math]\displaystyle{ \ln x =y }[/math]. This shows that the inverse function of [math]\displaystyle{ y=e^x }[/math] is [math]\displaystyle{ y=\ln x }[/math].

Not all functions have inverse functions: for example, function [math]\displaystyle{ f(x) = |x| }[/math] has none (because [math]\displaystyle{ |-1| = 1 = |1| }[/math], and [math]\displaystyle{ f^{-1}(x) }[/math] cannot be both 1 and -1), but every binary relation has its own inverse relation.

In some cases, finding the inverse of a function can be very difficult to do.

Inverse Function Media

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

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

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  The inverse of g ∘ f is f −1 ∘ g −1.

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  The graphs of y = f(x) and y = f −1(x). The dotted line is y = x.

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  The square root of x is a partial inverse to f(x) = x2.

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  The inverse of this cubic function has three branches.

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  The arcsine is a partial inverse of the sine function.

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  Example of right inverse with non-injective, surjective function

Related pages

• Inverse element
• Inverse triangle function
• Inverse hyperbolic function
• Invertible matrix
• Reciprocal