Algebra and Trigonometry 10th Edition

Published by Cengage Learning
ISBN 10: 9781337271172
ISBN 13: 978-1-33727-117-2

Chapter 2 - 2.6 - Combinations of Functions: Composite Functions - 2.6 Exercises - Page 219: 12


a) $\dfrac{2x^2+x-2}{x(x^2-1)}$ b) $\dfrac{2x^2-x-2}{x(x^2-1)}$ c) $\dfrac{2}{x(x^2-1)}$ d) $\dfrac{2(x^2-1)}{x}$ Domain: $(-\infty,-1)\cup(-1,0)\cup(0,1)\cup(1,\infty)$

Work Step by Step

We are given the functions: $f(x)=\dfrac{2}{x}$ $g(x)=\dfrac{1}{x^2-1}$ a) Determine $(f+g)(x)$: $(f+g)(x)=f(x)+g)(x)=\dfrac{2}{x}+\dfrac{1}{x^2-1}=\dfrac{2x^2+x-2}{x(x^2-1)}$ b) Determine $(f-g)(x)$: $(f-g)(x)=f(x)-g)(x)=\dfrac{2}{x}-\dfrac{1}{x^2-1}=\dfrac{2x^2-x-2}{x(x^2-1)}$ c) Determine $(fg)(x)$: $(fg)(x)=f(x)g)(x)=\dfrac{2}{x}\cdot\dfrac{1}{x^2-1}=\dfrac{2}{x(x^2-1)}$ d) Determine $\left(\dfrac{f}{g}\right)(x)$: $\left(\dfrac{f}{g}\right)(x)=\dfrac{\dfrac{2}{x}}{\dfrac{1}{x^2-1}}=\dfrac{2(x^2-1)}{x}$ Determine the domain of $\dfrac{f}{g}$: $x^2-1=0\Rightarrow x=\pm 1$ $x=0$ The domain is: $(-\infty,-1)\cup(-1,0)\cup(0,1)\cup(1,\infty)$
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