## Intermediate Algebra: Connecting Concepts through Application

$\text{a) } (f\circ g)(-3)=29.08 \\\\\text{b) } (g\circ f)(-3)=47$
$\bf{\text{Solution Outline:}}$ With \begin{array}{l}\require{cancel} f(x)= -0.6x-3.2 \\g(x)= -7x-4.8 ,\end{array} to find $(f\circ g)(7) ,$ find first $g(7) .$ Then substitute the result in $f.$ To find $(g\circ f)(7) ,$ find first $f(7) .$ Then substitute the result in $g.$ $\bf{\text{Solution Details:}}$ a) Since $(f\circ g)(7)=f(g(7)),$ find first $g(7).$ That is \begin{array}{l}\require{cancel} g(x)=-7x-4.8 \\\\ g(7)=-7(7)-4.8 \\\\ g(7)=-49-4.8 \\\\ g(7)=-53.8 .\end{array} Replacing $x$ with the result above in $f$ results to \begin{array}{l}\require{cancel} f(x)=-0.6x-3.2 \\\\ f(-53.8)=-0.6(-53.8)-3.2 \\\\ f(-53.8)=32.28-3.2 \\\\ f(-53.8)=29.08 .\end{array} Hence, $(f\circ g)(7)=f(g(7))=29.08 .$ b) Since $(g\circ f)(7)=g(f(7)),$ find first $f(7).$ Replacing $x$ with $7$ in $f$ results to \begin{array}{l}\require{cancel} f(x)=-0.6x-3.2 \\\\ f(7)=-0.6(7)-3.2 \\\\ f(7)=-4.2-3.2 \\\\ f(7)=-7.4 .\end{array} Replacing $x$ with the result above in $g$ results to \begin{array}{l}\require{cancel} g(x)=-7x-4.8 \\\\ g(-7.4)=-7(-7.4)-4.8 \\\\ g(-7.4)=51.8-4.8 \\\\ g(-7.4)=47 .\end{array} Hence, $(g\circ f)(7)=g(f(7))=47 .$ Therefore, \begin{array}{l}\require{cancel} \text{a) } (f\circ g)(-3)=29.08 \\\\\text{b) } (g\circ f)(-3)=47 .\end{array}