Answer
$\text{a) }
(f\circ g)(-3)=-0.5096
\\\\\text{b) }
(g\circ f)(-3)=7.6324$
Work Step by Step
$\bf{\text{Solution Outline:}}$
With
\begin{array}{l}\require{cancel}
f(x)=
0.68x+2.36
\\g(x)=
3.57x+6.49
,\end{array}
to find $
(f\circ g)(-3)
,$ find first $
g(-3)
.$ Then substitute the result in $f.$
To find $
(g\circ f)(-3)
,$ find first $
f(-3)
.$ Then substitute the result in $g.$
$\bf{\text{Solution Details:}}$
a) Since $(f\circ g)(-3)=f(g(-3)),$ find first $g(-3).$ That is
\begin{array}{l}\require{cancel}
g(x)=3.57x+6.49
\\\\
g(-3)=3.57(-3)+6.49
\\\\
g(-3)=-10.71+6.49
\\\\
g(-3)=-4.22
.\end{array}
Replacing $x$ with the result above in $f$ results to
\begin{array}{l}\require{cancel}
f(x)=0.68x+2.36
\\\\
f(-4.22)=0.68(-4.22)+2.36
\\\\
f(-4.22)=-2.8696+2.36
\\\\
f(-4.22)=-0.5096
.\end{array}
Hence, $
(f\circ g)(-3)=f(g(-3))=-0.5096
.$
b) Since $(g\circ f)(-3)=g(f(-3)),$ find first $f(-3).$ Replacing $x$ with $
-3
$ in $f$ results to
\begin{array}{l}\require{cancel}
f(x)=0.68x+2.36
\\\\
f(-3)=0.68(-3)+2.36
\\\\
f(-3)=-2.04+2.36
\\\\
f(-3)=0.32
.\end{array}
Replacing $x$ with the result above in $g$ results to
\begin{array}{l}\require{cancel}
g(x)=3.57x+6.49
\\\\
g(0.32)=3.57(0.32)+6.49
\\\\
g(0.32)=1.1424+6.49
\\\\
g(0.32)=7.6324
.\end{array}
Hence, $
(g\circ f)(-3)=g(f(-3))=7.6324
.$
Therefore,
\begin{array}{l}\require{cancel}
\text{a) }
(f\circ g)(-3)=-0.5096
\\\\\text{b) }
(g\circ f)(-3)=7.6324
.\end{array}
