Answer
f∘g $\frac {2x+4}{x}$ Domain: (-∞, -2) U (-2,0) U (0, ∞)
g∘f $\frac{1}{1+x}$Domain: (-∞, -1) U (-1,0) U (0, ∞)
f∘f $x$ Domain: (- ∞, 0) U (0, ∞)
g∘g $\frac{x}{3x+4}$ Domain: (-∞, -2) U (-2,-4/3) U (-4/3, ∞)
Work Step by Step
$f(x) = \frac{2}{x}$
$g(x) = \frac{x}{x+2} $
f∘g $f(\frac{x}{x+2}) = \frac {2} {\frac {x}{x+2}}$
= $\frac {2x+4}{x}$ Domain: (-∞, -2) U (-2,0) U (0, ∞)
g∘f $g(\frac{2}{x}) = \frac{\frac{2}{x}}{\frac{2}{x} + 2}$
= $\frac {2}{2+2x} = \frac{1}{1+x} $Domain: (-∞, -1) U (-1,0) U (0, ∞)
f∘f $f(\frac{2}{x}) = \frac{2}{\frac{2}{x}}$
=$x$ Domain: (- ∞, 0) U (0, ∞)
g∘g $g(\frac{x}{x+2}) = \frac{\frac{x}{x+2}}{\frac{x}{x+2} +2}$
=$\frac{x}{x +2(x+2)}$
=$\frac{x}{3x+4}$ Domain: (-∞, -2) U (-2,-4/3) U (-4/3, ∞)
