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: $-2\le x\le 2$
Range: $-4\le y\le 0$