Inverse function

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the inverse function of a <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> f (also called the inverse of f) is a <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> that undoes the operation of f. The inverse of f exists <a href="/facts/If_and_only_if/bYSxGJ66">if and only if</a> f is <a href="/facts/Bijection/j4gTuTmW">bijective</a>, and if it exists, is denoted by 
 
 
 
 
 f
 
 −
 1
 
 
 .
 
 
 {\displaystyle f^{-1}.}

For a function 
 
 
 
 f
 :
 X
 →
 Y
 
 
 {\displaystyle f\colon X\to Y}
 
, its inverse 
 
 
 
 
 f
 
 −
 1
 
 
 :
 Y
 →
 X
 
 
 {\displaystyle f^{-1}\colon Y\to X}
 
 admits an explicit description: it sends each element 
 
 
 
 y
 ∈
 Y
 
 
 {\displaystyle y\in Y}
 
 to the unique element 
 
 
 
 x
 ∈
 X
 
 
 {\displaystyle x\in X}
 
 such that f(x) = y.
As an example, consider the <a href="/facts/Real_number/R02gw5Pb">real-valued</a> function of a real variable given by f(x) = 5x − 7. One can think of f as the function which multiplies its input by 5 then subtracts 7 from the result. To undo this, one adds 7 to the input, then divides the result by 5. Therefore, the inverse of f is the function 
 
 
 
 
 f
 
 −
 1
 
 
 :
 
 R
 
 →
 
 R
 
 
 
 {\displaystyle f^{-1}\colon \mathbb {R} \to \mathbb {R} }
 
 defined by 
 
 
 
 
 f
 
 −
 1
 
 
 (
 y
 )
 =
 
 
 
 y
 +
 7
 
 5
 
 
 .
 
 
 {\displaystyle f^{-1}(y)={\frac {y+7}{5}}.}

Inverse function open-in-new

Inverse function