JEE Main & Advanced Mathematics Functions Inverse Function

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.