inverse function

English

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inverse function

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 ${\displaystyle f:X\rightarrow Y}$, 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, 2nd edition, Wiley, 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)

Related terms

• invertible

Translations

Further reading

• Bijection on Wikipedia.Wikipedia
• Inverse function theorem on Wikipedia.Wikipedia
• Inverse function on Encyclopedia of Mathematics
• Inverse Function on Wolfram MathWorld