Answer
a) $6x+3$; $D=(-\infty,\infty)$
b) $6x+9$; $D=(-\infty,\infty)$
c) $4x+9$; $D=(-\infty,\infty)$
d) $9x$; $D=(-\infty,\infty)$
Work Step by Step
We are given the functions:
$f(x)=2x+3$
$g(x)=3x$
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 $D_{f\circ g}$:
$(f\circ g)(x)=f(g(x))=f(3x)=2(3x)+3=6x+3$
$D_{f\circ g}=(-\infty,\infty)$
b) Determine $(g\circ f$) and its domain $D_{g\circ f}$:
$(g\circ f)(x)=g(f(x))=g(2x+3)=3(2x+3)=6x+9$
$D_{g\circ f}=(-\infty,\infty)$
c) Determine $(f\circ f$) and its domain $D_{f\circ f}$:
$(f\circ f)(x)=f(f(x))=f(2x+3)=2(2x+3)+3=4x+9$
$D_{f\circ f}=(-\infty,\infty)$
d) Determine $(g\circ g$) and its domain $D_{g\circ g}$:
$(g\circ g)(x)=g(g(x))=g(3x)=3(3x)=9x$
a) $6x+3$; $D=(-\infty,\infty)$
b) $6x+9$; $D=(-\infty,\infty)$
c) $4x+9$; $D=(-\infty,\infty)$
d) $9x$; $D=(-\infty,\infty)$
$D_{g\circ g}=(-\infty,\infty)$
