# Chapter 3 - Exponents, Polynomials and Functions - 3.3 Composing Functions - 3.3 Exercises - Page 258: 23

$\text{a) } f(g(x))=3x^2+12x+35 \\\\\text{b) } g(f(x))=9x^2+42x+55$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$ With \begin{array}{l}\require{cancel} f(x)= 3x+5 \\g(x)= x^2+4x+10 ,\end{array} replace $x$ with $g(x)$ in $f$ to find $f(g(x)).$ To find $g(f(x)),$ replace $x$ with $f(x)$ in $g.$ $\bf{\text{Solution Details:}}$ Replacing $x$ with $g(x)$ in $f,$ then \begin{array}{l}\require{cancel} f(g(x))=f(x^2+4x+10) \\\\ f(g(x))=3(x^2+4x+10)+5 \\\\ f(g(x))=3x^2+12x+30+5 \\\\ f(g(x))=3x^2+12x+35 .\end{array} Replacing $x$ with $f(x)$ in $g.$ Hence, \begin{array}{l}\require{cancel} g(f(x))=g(3x+5) \\\\ g(f(x))=(3x+5)^2+4(3x+5)+10 \\\\ g(f(x))=(9x^2+30x+25)+(12x+20)+10 \\\\ g(f(x))=9x^2+42x+55 .\end{array} Hence, \begin{array}{l}\require{cancel} \text{a) } f(g(x))=3x^2+12x+35 \\\\\text{b) } g(f(x))=9x^2+42x+55 .\end{array}

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