Answer
a) 1; b) $x^{3/2}$; c) $(\sin\sqrt x)^3$
d) $\mathbb{R}$; e) $[-1,1]$
Work Step by Step
We are given the functions:
$f(x)=x^3$
$g(x)=\sin x$
$h(x)=\sqrt x$
a) Evaluate $h(g(\pi/2))$:
$h(g(\pi/2))=\sqrt {\sin (\pi/2)}=\sqrt 1=1$
b) Find $h(f(x))$:
$h(f(x))=\sqrt(x^3)=x^{3/2}$
c) Find $f(g(h(x)))$:
$f(g(h(x)))=f(g(\sqrt x)=f(\sin\sqrt x)=(\sin\sqrt x)^3$
d) Find $(g\circ f)(x)$:
$(g\circ f)(x)=g(f(x))=g(x^3)=\sin x^3$
The domain of the function is the set of all real numbers $\mathbb{R}$.
e) Find $(f\circ g)(x)$:
$(f\circ g)(x)=f(g(x))=f(\sin x)=(\sin x^3)^3$
We have:
$-1\leq \sin x\leq 1$
$-1\leq (\sin x)^3\leq 1$
The range is $[-1,1]$.
