Answer
Given $f(x) = \sqrt x$ and $g(x)= \frac {2} {x-4}$, we need to find the following and their respective domains:
a. $\sqrt {\frac{2}{x-4}}$
Domain: (4, ∞)
b.$\frac { 2}{\sqrt x - 4}$
Domain: [0, 16) U (16, ∞)
c. $\sqrt {\sqrt x}$
Domain: [0, ∞)
d. $\frac{x-4}{9-2x}$
Domain: (-∞, 4) U (4, 9/2) U (9/2, ∞)
Work Step by Step
Given $f(x) = \sqrt x$ and $g(x)= \frac {2} {x-4}$, we need to find the following and their respective domains:
a. f o g
$f(\frac{2}{x-4}) = \sqrt {\frac{2}{x-4}}$
Domain: (4, ∞)
b. g o f
$g(\sqrt x) = \frac { 2}{\sqrt x - 4}$
Domain: [0, 16) U (16, ∞)
(since the denominator can not equal zero, so 16 is excluded)
c. f o f
$f ( \sqrt x) = \sqrt {\sqrt x}$
Domain: [0, ∞)
d. g o g
$g (\frac {2}{x-4}) = \frac{2}{\frac {2}{x-4} - 4} = \frac { 2(x-4)}{2 - 4(x-4)} = \frac{2x-8}{18-4x} $
$ = \frac{x-4}{9-2x}$
Domain: (-∞, 4) U (4, 9/2) U (9/2, ∞)