# inverse function

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Wikipedia

### Noun

inverse function (plural inverse functions)

1. (mathematics) For a given function f, another function, denoted f−1, that reverses the mapping action of f; (formally) given a function $f:X\rightarrow Y$ , a function $g:Y\rightarrow X$ such that, $\forall x\in X,\ f(x)=y\implies g(y)=x$ .
Halving is the inverse function of doubling.
If an inverse function exists for a given function, then it is unique.
The inverse function of an inverse function is the original function.
• 1995, Nicholas M. Karayanakis, Advanced System Modelling and Simulation with Block Diagram Languages, CRC Press, page 217,
In the context of linearization, we recall the reflective property of inverse functions; the ƒ curve contains the point (a,b) if and only if the ƒ -1 curve contains the point (b,a).
• 2014, Mary Jane Sterling, Trigonometry For Dummies, Wiley, 2nd Edition, page 51,
An example of another function that has an inverse function is $f(x)=4x+5$ .
Its inverse is $f^{-1}(x)={\frac {x-5}{4}}$ .
• 2014, Mark Ryan, Calculus For Dummies, Wiley, 2nd Edition, page 147,
If $f$ and $g$ are inverse functions, then
$f'(x)={\frac {1}{g'(f(x))}}$ In words, this formula says that the derivative of a function, $f$ , with respect to $x$ , is the reciprocal of the derivative of its inverse function with respect to $f$ .

#### Synonyms

• (function that reverses the mapping action of a given function): anti-function (obsolete or nonstandard in this sense)