inverse function

English

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, 2nd edition, Wiley, 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)

Related terms

• invertible

Translations

function that reverses the mapping action of a given function

• Catalan: funció inversa f
• Czech: inverzní funkce f
• Danish: invers funktion c, omvendt funktion c
• Finnish: käänteisfunktio
• French: fonction réciproque f
• German: Umkehrfunktion (de) f, inverse Funktion f
• Hungarian: inverz függvény (hu)
• Italian: funzione inversa (it) f
• Japanese: 逆関数 (ja) (gyaku-kansū)
• Korean: 역함수 (ko) (yeokhamsu)
• Latin: fūnctiō inversa f
• Norwegian:

  Bokmål: invers funksjon m

  Nynorsk: invers funksjon m

• Polish: funkcja odwrotna (pl) f
• Portuguese: função inversa f
• Russian: обратная фу́нкция f (obratnaja fúnkcija)
• Serbo-Croatian:

  Cyrillic: инверзна функција f

  Roman: inverzna funkcija f

• Slovak: inverzná funkcia f
• Slovene: inverzna funkcija f
• Spanish: función recíproca f, función inversa f
• Swedish: invers funktion c
• Tagalog: baligtad na kabisa
• Ukrainian: обернена функція f (obernena funkcija)

Further reading

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