Answer
a) $f\circ g =\dfrac{3x}{2-x}$
Domain: $\{x|x\ne0,x\ne1,x\ne2 \}$
b) $g\circ f =\dfrac{2x-2}{3}$
Domain: $\{x|x\ne0,x\ne1 \}$
c) $f\circ f =\dfrac{3x-3}{4-x}$
Domain: $\{x|x\ne1,x\ne4 \}$
d) $g\circ g = x$
Domain: $\{x|x\ne0\}$
Work Step by Step
a) $f\circ g = f(g(x)) = $
$\dfrac{3}{\frac{2}{x}-1}=$
$\dfrac{3}{\frac{2}{x}-\frac{x}{x}}=$
$\dfrac{3}{\frac{2-x}{x}}=$
$\dfrac{3x}{2-x}$
Domain: $\{x|x\ne0,x\ne1,x\ne2 \}$
b) $g\circ f = g(f(x)) =$
$\dfrac{2}{\frac{3}{x-1}}=$
$\dfrac{2(x-1)}{3}=$
$\dfrac{2x-2}{3}$
Domain: $\{x|x\ne0,x\ne1 \}$
c) $f\circ f = f(f(x))=$
$\dfrac{3}{\frac{3}{x-1}-1}=$
$\dfrac{3}{\frac{3}{x-1}-\frac{x-1}{x-1}}=$
$\dfrac{3}{\frac{3-(x-1)}{x-1}}=$
$\dfrac{3(x-1)}{3-x+1}=$
$\dfrac{3x-3}{4-x}$
Domain: $\{x|x\ne1,x\ne4 \}$
d) $g\circ g = g(g(x))=$
$\dfrac{2}{\frac{2}{x}}=$
$\dfrac{2x}{2}=$
$x$
Domain: $\{x|x\ne0\}$
