# Chapter 3 - Exponents, Polynomials and Functions - 3.3 Composing Functions - 3.3 Exercises - Page 258: 25

$\text{a) } f(g(x))=4x^2+x+3 \\\\\text{b) } g(f(x))=4x^2+17x+19$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$ With \begin{array}{l}\require{cancel} f(x)= x+2 \\g(x)= 4x^2+x+1 ,\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(4x^2+x+1) \\\\ f(g(x))=(4x^2+x+1)+2 \\\\ f(g(x))=4x^2+x+3 .\end{array} Replacing $x$ with $f(x)$ in $g.$ Hence, \begin{array}{l}\require{cancel} g(f(x))=g(x+2) \\\\ g(f(x))=4(x+2)^2+(x+2)+1 \\\\ g(f(x))=4(x^2+4x+4)+(x+2)+1 \\\\ g(f(x))=(4x^2+16x+16)+(x+2)+1 \\\\ g(f(x))=4x^2+17x+19 .\end{array} Hence, \begin{array}{l}\require{cancel} \text{a) } f(g(x))=4x^2+x+3 \\\\\text{b) } g(f(x))=4x^2+17x+19 .\end{array}

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