**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.

*play_arrow*Some Important Definitions*play_arrow*Intervals*play_arrow*Definition of Function*play_arrow*Domain, Co-domain and Range of Function*play_arrow*Algebra of Functions*play_arrow*Kinds of function*play_arrow*Even and Odd Function*play_arrow*Periodic Function*play_arrow*Composite Function*play_arrow*Inverse Function*play_arrow*Limit of a Function*play_arrow*Fundamental Theorems on Limits*play_arrow*Methods of Evaluation of Limits*play_arrow*Introduction*play_arrow*Continuity of a Function at a Point*play_arrow*Continuity From Left and Right*play_arrow*Discontinuous Function*play_arrow*Differentiability of a Function at a Point

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