Answer
(a) $\dfrac{11}{6}$
(b) $\dfrac{3}{2}$
(c) $1$
(d) $\dfrac{12}{17}$
Work Step by Step
(a) $(f\circ g)(4)= f(g(4))=f\left(\dfrac{3}{4^{2}+2}\right)=f\left(\dfrac{1}{6}\right)=\left|\dfrac{1}{6}-2\right|=\left|-\dfrac{11}{6}\right|=\dfrac{11}{6}$
(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) $(g\circ g)(0)=g(g(0))=g\left(\dfrac{3}{0^{2}+2}\right)=g\left(\dfrac{3}{2}\right)=\dfrac{3}{\left(\frac{3}{2}\right)^{2}+2}=\dfrac{3}{\frac{9}{4}+\frac{8}{4}}=\dfrac{3}{\frac{17}{4}}=3 \cdot \dfrac{4}{17}=\dfrac{12}{17}$
