Answer
$\text{a) }
(f\circ g)(x)=-56x+116
\\\text{b) }
(g\circ f)(x)=-56x-73$
Work Step by Step
$\bf{\text{Solution Outline:}}$
With
\begin{array}{l}\require{cancel}
f(x)=
7x+11
\\g(x)=
-8x+15
,\end{array}
use the definition of function composition to find $
(f\circ g)(x)
,$ and $(g\circ f)(x).$
$\bf{\text{Solution Details:}}$
Using $(f\circ g)(x)=f(g(x)),$ then replace $x$ with $g(x)$ in $f$. Hence,
\begin{array}{l}\require{cancel}
(f\circ g)(x)=f(g(x))
\\\\
(f\circ g)(x)=f(-8x+15)
\\\\
(f\circ g)(x)=7(-8x+15)+11
\\\\
(f\circ g)(x)=-56x+105+11
\\\\
(f\circ g)(x)=-56x+116
.\end{array}
Using $(g\circ f)(x) =g(f(x)),$ then replace $x$ with $f(x)$ in $g.$ Hence,
\begin{array}{l}\require{cancel}
(g\circ f)(x)=g(f(x))
\\\\
(g\circ f)(x) =g(7x+11)
\\\\
(g\circ f)(x) =-8(7x+11)+15
\\\\
(g\circ f)(x) =-56x-88+15
\\\\
(g\circ f)(x) =-56x-73
.\end{array}
Hence,
\begin{array}{l}\require{cancel}
\text{a) }
(f\circ g)(x)=-56x+116
\\\text{b) }
(g\circ f)(x)=-56x-73
.\end{array}
