College Algebra 7th Edition

Published by Brooks Cole
ISBN 10: 1305115546
ISBN 13: 978-1-30511-554-5

Chapter 2, Functions - Section 2.7 - Combining Functions - 2.7 Exercises - Page 252: 12

Answer

For $f+g, f-g, fg$: $D=\{x|x\in\mathbb{R}\}$, For $\frac{f}{g}$: $D_{\frac{f}{g}}=\left\{x|x\in\mathbb{R}-\left\{ \pm \frac{1}{\sqrt 3}\right\}\right\}$

Work Step by Step

$f(x)=x^2+2x$, $D_f=\{x|x\in\mathbb{R}\}$, $g(x)=3x^2-1$, $D_g=\{x|x\in\mathbb{R}\}$, thus, $f+g=x^2+2x+3x^2-1=4x^2+2x-1$, $D_{f+g}=\{x|x\in\mathbb{R}\}$, $f-g=x^2+2x-3x^2+1=-2x^2+2x+1$, $D_{f-g}=\{x|x\in\mathbb{R}\}$, $f \times g=(x^2+2x)(3x^2-1)=3x^4-x^2+6x^3-2x\\ =3x^4+6x^3-x^2-2x$ $D_{f \times g}=\{x|x\in\mathbb{R}\}$, $\frac{f}{g}=\frac{x^2+2x}{3x^2-1}$, $3x^2-1=0\Rightarrow x=\pm\dfrac{1}{\sqrt 3}$ $D_{\frac{f}{g}}=\left\{x|x\in\mathbb{R}-\left\{ \pm \frac{1}{\sqrt 3}\right\}\right\}$
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