## Intermediate Algebra: Connecting Concepts through Application

$\text{a) } (f\circ g)(x)=-56x+116 \\\text{b) } (g\circ f)(x)=-56x-73$
$\bf{\text{Solution Outline:}}$ 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}