Question

For the following problems, find the function f ° g and the domain of f ° g .

2) f(x) =
$\frac{x}{x + 4}$
;
$g(x) = \frac{16}{x + 2}$


3) f(x) =
$\sqrt{x}$
; g(x)= 5x+5

Answer 1

$2)\\f(x)=\dfrac{x}{x+4}\\\\g(x)=\dfrac{16}{x+2}\\\\Domain:\\\\D_f:x\neq-4\\\\D_g:x\neq-2\\\\(f\ \circ\ g)(x)=\dfrac{\frac{x}{x+4}}{\frac{16}{x+2}}=\dfrac{x}{x+4}\cdot\dfrac{x+2}{16}=\dfrac{x^2+2x}{16x+64}\\\\Domain:\\\\16x+64\neq0\to16x\neq-64\to\boxed{x\neq-4}$

$3)\\f(x)=\sqrt{x}\\\\g(x)=5x+5\\\\Domain:\\\\D_f:x\geq0\\\\D_g:x\in\mathbb{R}\\\\(f\ \circ\ g)(x)=\sqrt{5x+5}\\\\Domain:\\\\5x+5\geq0\to5x\geq-5\to\boxed{x\geq-1}$