# JEE Main & Advanced Mathematics Functions Inverse Function

## Inverse Function

Category : JEE Main & Advanced

If $f:A\to B$ be a one-one onto (bijection) function, then the mapping ${{f}^{-1}}:B\to A$ which associates each element $b\in B$ with element $a\in A,$ such that $f(a)=b,$ is called the inverse function of the function $f:A\to B$.

${{f}^{-1}}:B\to A,\,\,{{f}^{-1}}(b)=a\Rightarrow f(a)=b$

In terms of ordered pairs inverse function is defined as ${{f}^{-1}}=(b,\,a)$ if $(a,\,\,b)\in f$.

For the existence of inverse function, it should be one-one and onto.

Properties of Inverse function :

(1) Inverse of a bijection is also a bijection function.

(2) Inverse of a bijection is unique.

(3) ${{({{f}^{-1}})}^{-1}}=f$

(4) If $f$ and $g$ are two bijections such that $(gof)$ exists then ${{(gof)}^{-1}}={{f}^{-1}}o{{g}^{-1}}$.

(5) If $f:A\to B$ is a bijection then ${{f}^{-1}}\,.\,B\to A$ is an inverse function of $f.\,{{f}^{-1}}$ $of={{l}_{A}}$ and $fo{{f}^{-1}}={{l}_{B}}$. Here ${{l}_{A}},$ is an identity function on set A, and ${{l}_{B}},$ is an identity function on set B.

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