# JEE Main & Advanced Mathematics Functions Composite Function

## Composite Function

Category : JEE Main & Advanced

• 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.

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