Algebra/Chapter 11/Composite Functions

Composite function

A composite function $h$ can be defined as the composite of the two functions $f$ and $g$ and denoted as $h(x)=f(g(x))$ (read h of x is equal to f of g of x) or $h(x)=(f\circ g)(x)$.

Example:

Let $f(x)=2x+1$     $g(x)=5x-3$
 $h(x)=f(g(x))$
 $h(x)=f(5x-3)$
 $h(x)=2(5x-3)+1$
 $h(x)=10x-6+1$
∴$h(x)=10x-5$

Example:

Let $f(x)=-{\sqrt {16-x}}\,$     $g(x)=4x^{2}\,$
$(f\circ g)(x)=f(4x^{2})\,$
$(f\circ g)(x)=-{\sqrt {16-(4x^{2})}}\,$
$(f\circ g)(x)=-{\sqrt {4(4-x^{2})}}\,$
$(f\circ g)(x)=-{\sqrt {4}}{\sqrt {(4-x^{2})}}\,$
$(f\circ g)(x)=-2{\sqrt {4-x^{2}}}\,$
Domain: $-2\leq x\leq 2$
Range: $-4\leq y\leq 0$