Inverse and Composite Functions

Inverse and Composite Functions
Composite functions:
    Let A, B, & C are three non-empty sets and let  \(f:A \to B,g:B \to C\)be two functions.Since ‘f’ is a function from A to B, therefore for each  \(x\in A\)there exists a unique element \(f(x) \in B\).  Again since g is a function from B to C, therefore corresponding to \(f(x) \in B\), there exists a unique element  \(g[f(x)] \in C\).  Thus for each, there exists a unique element.
    It follows from the above discussion that f and g when considered together define a new function from A to C.  This function  is called the composition of f and g and is denoted by gof.

Definition : 
    Let \(f:A \to B\)  and \(g:B \to C\) be two functions.  Then a function \(gof:A \to C\) defined by \((gof)(x) = g[f(x)]\) , for all \(x\in A\) is called the composition of f and g.
Note :(1)     For the composition of “ gof ” to exist, the range of ‘f’ must be the subset of the domain of ‘g’.
(2) For “fog” exists if range of g is a subset of domain of f.     (3) \(fog \ne gof\)
(4)\(f:A \to B \Rightarrow {f^{ - 1}}:B \to A\)
\(g:B \to C \Rightarrow {g^{ - 1}}:C \to B\)
\(\Rightarrow {f^{ - 1}}o{g^{ - 1}}:C \to A\) and \(gof:A \to C\)
\( \Rightarrow {f^{ - 1}}o{g^{ - 1}}:C \to A\) and \({(gof)^{ - 1}}:C \to A\)
\(\therefore \,\,{(gof)^{ - 1}} = {f^{ - 1}}o{g^{ - 1}}\)