Answer
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Work Step by Step
$f(x)=\frac{x+1}{x-1}, g(x)=\frac{1}{x}$
$(f \circ g) (x)=\frac{\frac{1}{x}+1}{\frac{1}{x}-1}=\frac{1+x}{1-x},$
Domain of $(f\circ g)(x)$ is $x \in \mathbb{R} \ne \{0,1\}$
$(g\circ f)(x)=\frac{1}{\frac{x+1}{x-1}}=\frac{x-1}{x+1},$
Domain of $(f\circ g )(x)$ is $x\in \mathbb{R} \ne \{-1, 1\}$
$(f\circ f)(x)=\frac{\frac{x+1}{x-1}+1}{\frac{x+1}{x-1}-1}=x,$
Domain of $(f\circ f )(x)$ is $x\in \mathbb{R} \ne 1$
$(g\circ g)(x)=\frac{1}{\frac{1}{x}}=x,$
Domain of $(g\circ g)(x)$ is $x\in \mathbb{R} \ne 0$
