Algebra/Chapter 11/Composite Functions

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

Example:

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

Example:

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