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: 11


a) $\dfrac{x^4+x^3+x}{x+1}$ b) $\dfrac{-x^4-x^3+x}{x+1}$ c) $\dfrac{x^4}{x+1}$ d) $\dfrac{1}{x^2(x+1)}$ Domain: $(-\infty,-1)\cup(-1,0)\cup(0,\infty)$

Work Step by Step

We are given the functions: $f(x)=\dfrac{x}{x+1}$ $g(x)=x^3$ a) Determine $(f+g)(x)$: $(f+g)(x)=f(x)+g)(x)=\dfrac{x}{x+1}+x^3=\dfrac{x^4+x^3+x}{x+1}$ b) Determine $(f-g)(x)$: $(f-g)(x)=f(x)-g)(x)=\dfrac{x}{x+1}-x^3=\dfrac{-x^4-x^3+x}{x+1}$ c) Determine $(fg)(x)$: $(fg)(x)=f(x)g)(x)=\dfrac{x}{x+1}\cdot x^3=\dfrac{x^4}{x+1}$ d) Determine $\left(\dfrac{f}{g}\right)(x)$: $\left(\dfrac{f}{g}\right)(x)=\dfrac{\dfrac{x}{x+1}}{x^3}=\dfrac{1}{x^2(x+1)}$ Determine the domain of $\dfrac{f}{g}$: $x^2(x+1)=0$ $x=0$ or $x=-1$ The domain is: $(-\infty,-1)\cup(-1,0)\cup(0,\infty)$
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