inverse function
التعريفات والمعاني
== English ==
=== Pronunciation ===
=== Noun ===
inverse function (plural inverse functions)
(mathematics) For a given function f, another function, denoted f−1, that reverses the mapping action of f; (formally) given a function
f
:
X
→
Y
{\displaystyle f:X\rightarrow Y}
, a function
g
:
Y
→
X
{\displaystyle g:Y\rightarrow X}
such that,
∀
x
∈
X
,
f
(
x
)
=
y
⟹
g
(
y
)
=
x
{\displaystyle \forall x\in X,\ f(x)=y\implies g(y)=x}
.
2014, Mark Ryan, Calculus For Dummies, Wiley, 2nd Edition, page 147,
If
f
{\displaystyle f}
and
g
{\displaystyle g}
are inverse functions, then
f
′
(
x
)
=
1
g
′
(
f
(
x
)
)
{\displaystyle f'(x)={\frac {1}{g'(f(x))}}}
In words, this formula says that the derivative of a function,
f
{\displaystyle f}
, with respect to
x
{\displaystyle x}
, is the reciprocal of the derivative of its inverse function with respect to
f
{\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