# inverse function

## English

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### 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 ${\displaystyle f:X\rightarrow Y}$, a function ${\displaystyle g:Y\rightarrow X}$ such that, ${\displaystyle \forall x\in X,\ f(x)=y\implies g(y)=x}$.
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 ${\displaystyle f(x)=4x+5}$.
Its inverse is ${\displaystyle f^{-1}(x)={\frac {x-5}{4}}}$.
• 2014, Mark Ryan, Calculus For Dummies, Wiley, 2nd Edition, page 147,
If ${\displaystyle f}$ and ${\displaystyle g}$ are inverse functions, then
${\displaystyle f'(x)={\frac {1}{g'(f(x))}}}$
In words, this formula says that the derivative of a function, ${\displaystyle f}$, with respect to ${\displaystyle x}$, is the reciprocal of the derivative of its inverse function with respect to ${\displaystyle f}$.

#### Synonyms

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