Answer
a) $\dfrac{11}{6}$; b) $\dfrac{3}{2}$; c) $1$; d) $\dfrac{12}{17}$
Work Step by Step
We are given the functions:
$f(x)=|x-2|$
$g(x)=\dfrac{3}{x^2+2}$
a) $(f\circ g)(4)=f(g(4))=f\left(\dfrac{3}{4^2+2}\right)=f\left(\dfrac{3}{18}\right)=f\left(\dfrac{1}{6}\right)=\left|\dfrac {1}{6}-2\right|=\left|-\dfrac {11}{6}\right|=\dfrac{11}{6}$
b) $(g\circ f)(2)=g(f(2))=g(|2-2|)=g(0)=\dfrac{3}{0^2+2}=\dfrac{3}{2}$
c) $(f\circ f)(1)=f(f(1))=f(|1-2|)=f(1)=|1-2|=1$
d) $(g\circ g)(0)=g(g(0))=g\left(\dfrac{3}{0^2+2}\right)=g\left(\dfrac{3}{2}\right)=\dfrac{3}{\left(\dfrac{3}{2}\right)^2+2}=\dfrac{3}{\dfrac{17}{4}}=\dfrac{12}{17}$
