JEE Main & Advanced Mathematics Functions Composite Function

• If \[f:A\to B\] and \[g:B\to C\] are two function then the composite function of \[f\] and \[g,\]

\[gof\,A\to C\] will be defined as \[gof(x)=g\,[f(x)],\,\forall x\in A\]

(1) Properties of composition of function :

(i) \[f\] is even, \[g\] is even \[\Rightarrow \]\[fog\] even function. 

(ii) \[f\] is odd, \[g\] is odd \[\Rightarrow \]\[fog\] is odd function.

(iii) \[f\] is even, \[g\] is odd   \[\Rightarrow \]\[fog\] is even function.           

(iv) \[f\] is odd, \[g\] is even \[\Rightarrow \]\[fog\] is even function.             

(v) Composite of functions is not commutative i.e., \[fog\,\ne \,gof\].

(vi) Composite of functions is associative i.e., \[(fog)oh\,=\,fo(goh)\]               

(vii) If \[f:A\to B\] is bijection and \[g:B\to A\] is inverse of  \[f\]. Then \[fog={{I}_{B}}\] and \[gof={{I}_{A}}.\]

where, \[{{I}_{A}}\] and \[{{I}_{B}}\] are identity functions on the sets A and B respectively.

(viii) If \[f:A\to B\] and \[g:B\to C\] are two bijections, then \[gof:A\to C\] is bijection and \[{{(gof)}^{-1}}=({{f}^{-1}}o{{g}^{-1}}).\]

(ix) \[fog\ne gof\] but if, \[fog=gof\] then either \[{{f}^{-1}}=g\] or \[{{g}^{-1}}=f\] also, \[(fog)\,(x)=(gof)\,(x)=(x).\]

(x) \[gof(x)\] is simply the g-image of \[f(x),\] where \[f(x)\] is f-image of elements \[x\in A\].

(xi) Function \[gof\] will exist only when range of \[f\] is the subset of domain of \[g\].

(xii) \[fog\] does not exist if range of g is not a subset of domain of \[f\].

(xiii) \[fog\] and \[gof\] may not be always defined.

(xiv) If both \[f\] and \[g\] are one-one, then \[fog\] and \[gof\] are also one-one.

(xv) If both \[f\] and \[g\] are onto, then \[gof\] is onto.