Answer
$\text{a) }
(f\circ g)(x)=-165x+471
\\\text{b) }
(g\circ f)(x) =-165x-274$
Work Step by Step
$\bf{\text{Solution Outline:}}$
With
\begin{array}{l}\require{cancel}
f(x)=
11x+21
\\g(x)=
-15x+41
,\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(-15x+41)
\\\\
(f\circ g)(x)=11(-15x+41)+21
\\\\
(f\circ g)(x)=-165x+451+21
\\\\
(f\circ g)(x)=-165x+471
.\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(11x+21)
\\\\
(g\circ f)(x) =-15(11x+21)+41
\\\\
(g\circ f)(x) =-165x-315+41
\\\\
(g\circ f)(x) =-165x-274
.\end{array}
Hence,
\begin{array}{l}\require{cancel}
\text{a) }
(f\circ g)(x)=-165x+471
\\\text{b) }
(g\circ f)(x) =-165x-274
.\end{array}
