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