# Composite Function

Let

*A*,*B,*and*C*be three non-empty sets.Let

*f*:*A*â†’*B*and*g*:*B*â†’*C*be two functions then*gof*:*A*â†’*C*. This function is called composition of*f*and*g*, given by*gof*(*x*) =*g*(*f*(*x*)) âˆ€*x*âˆˆ*A*.Thus the image of every

*x*âˆˆ*A*under the function*gof*is the*g*-image of the*f*-image of*x*.The

*gof*is defined only if âˆ€*x*âˆˆ*A*,*f*(*x*) is an element of the domain of*g*so that we can take its*g*-image.The range of

*f*must be a subset of the domain of*g*in*gof*.# Properties of composite functions

- The composition of function is not commutative, i.e.,
*fog*â‰*gof*. - The composition of function is associative, i.e., if
*h*:*A*â†’*B*, g:*B*â†’*C*and*f*:*C*â†’*D*be three functions, then (*fog*)*oh*=*fo*(*goh*). - The composition of any function with the identity function is the function itself, i.e.,
*f*:*A*â†’ B then*foI*=_{A}*I*=_{B}of*f*where*I*and_{A}*I*are the identity functions of_{B}*A*and*B*, respectively.

# Identical Function

Two functions

*f*and*g*are said to be identical if- The domain of
*f*= the domain of g, i.e.,*D*=_{f}*D*_{g} - The range of
*f*= the range of*g* *f*(*x*) =*g*(*x*) âˆ€ x âˆˆ*D*or_{f}*x*âˆˆ*D*e.g.,_{g},*f*(*x*) =*x*and*g*(*x*) = are not identical functions as*D*=_{f}*D*, but_{g}*R*=_{f}*R*,*R*= [0, âˆž)_{g}