Answer
a) $4x^4+12x^2+10$; $D=(-\infty,\infty)$
b) $2x^4+4x^2+5$; $D=(-\infty,\infty)$
c) $x^4+2x^2+2$; $D=(-\infty,\infty)$
d) $8x^4+24x^2+21$; $D=(-\infty,\infty)$
Work Step by Step
We are given the functions:
$f(x)=x^2+1$
$g(x)=2x^2+3$
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 $D_{f\circ g}$:
$(f\circ g)(x)=f(g(x))=f(2x^2+3)=(2x^2+3)^2+1=4x^4+12x^2+10$
$D_{f\circ g}=(-\infty,\infty)$
b) Determine $g\circ f$ and its domain $D_{g\circ f}$:
$(g\circ f)(x)=g(f(x))=g(x^2+1)=2(x^2+1)^2+3=2x^4+4x^2+5$
$D_{g\circ f}=(-\infty,\infty)$
c) Determine $f\circ f$ and its domain $D_{f\circ f}$:
$(f\circ f)(x)=f(f(x))=f(x^2+1)=(x^2+1)^2+1=x^4+2x^2+2$
$D_{f\circ f}=(-\infty,\infty)$
d) Determine $g\circ g$ and its domain $D_{g\circ g}$:
$(g\circ g)(x)=g(g(x))=g(2x^2+3)=2(2x^2+3)^2+3=8x^4+24x^2+21$
