Answer
a) $f(x)=\sin x$; $g(x)=x^2+1$
b) $f(x)=x^{-3}$; $g(x)=x^2-4$
c) $f(x)=e^x$; $g(x)=\cos 2x$
Work Step by Step
a) We are given the function:
$h(x)=\sin (x^2+1)$
Find the functions $f$ and $g$ so that $h=f\circ g$:
$g(x)=x^2+1$
$f(x)=\sin x$
Check:
$(f\circ g)(x)=f(g(x))=f(x^2+1)=\sin (x^2+1)$
b) We are given the function:
$h(x)=(x^2-4)^{-3}$
Find the functions $f$ and $g$ so that $h=f\circ g$:
$g(x)=x^2-4$
$f(x)=x^{-3}$
Check:
$(f\circ g)(x)=f(g(x))=f(x^2-4)=(x^2-4)^{-3}$
c) We are given the function:
$h(x)=e^{\cos 2x}$
Find the functions $f$ and $g$ so that $h=f\circ g$:
$g(x)=\cos 2x$
$f(x)=e^x$
Check:
$(f\circ g)(x)=f(g(x))=f(\cos 2x)=e^{\cos 2x}$
