Answer
a) $\dfrac{2\sqrt{10}}{25}$; b) $\dfrac{2}{2\sqrt 2+1}$; c) $1$; d) $\dfrac{2}{3}$.
Work Step by Step
We are given the functions:
$f(x)=x^{3/2}$
$g(x)=\dfrac{2}{x+1}$
a) $(f\circ g)(4)=f(g(4))=f\left(\dfrac{2}{4+1}\right)=f\left(\dfrac{2}{5}\right)=\left(\dfrac{2}{5}\right)^{3/2}=\dfrac{2}{5}\sqrt{\dfrac{2}{5}}=\dfrac{2\sqrt{10}}{25}$
b) $(g\circ f)(2)=g(f(2))=g\left(2^{3/2}\right)=g(2\sqrt 2)=\dfrac{2}{2\sqrt 2+1}$
c) $(f\circ f)(1)=f(f(1))=f\left(1^{3/2}\right)=f\left(1\right)=1^{3/2}=1$
d) $(g\circ g)(0)=g(g(0))=g\left(\dfrac{2}{0+1}\right)=g\left(2\right)=\dfrac{2}{2+1}=\dfrac{2}{3}$
