Answer
a) $x^4+8x^2+16$; $D=(-\infty,\infty)$
b) $x^4+4$; $D=(-\infty,\infty)$
c) $x^4$; $D=(-\infty,\infty)$
d) $x^4+8x^2+20$; $D=(-\infty,\infty)$
Work Step by Step
We are given the functions:
$f(x)=x^2$
$g(x)=x^2+4$
Let's note:
$D_f$=the domain of $f$
$D_g$=the domain of $g$
We have:
$D_f=(-\infty,\infty)$
$D_g=(-\infty,\infty)$
a) Determine $f\circ g$ and its domain $D_{f\circ g}$:
$(f\circ g)(x)=f(g(x))=f(x^2+4)=(x^2+4)^2=x^4+8x^2+16$
$D_{f\circ g}=(-\infty,\infty)$
b) Determine $g\circ f$ and its domain $D_{g\circ f}$:
$(g\circ f)(x)=g(f(x))=g(x^2)=(x^2)^2+4=x^4+4$
$D_{g\circ f}=(-\infty,\infty)$
c) Determine $f\circ f$ and its domain $D_{f\circ f}$:
$(f\circ f)(x)=f(f(x))=f(x^2)=(x^2)^2=x^4$
$D_{f\circ f}=(-\infty,\infty)$
d) Determine $g\circ g$ and its domain $D_{g\circ g}$:
$(g\circ g)(x)=g(g(x))=g(x^2+4)=(x^2+4)^2+4=x^4+8x^2+20$
