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: 8


a) $x^2+3x-15$ b) $-x^2+3x+17$ c) $3x^3+x^2-48x-16$ d) $\dfrac{3x+1}{x^2-16}$ Domain: $\left(-\infty,-4\right)\cup\left(-4,4\right)\cup\left(4,\infty\right)$

Work Step by Step

We are given the functions: $f(x)=3x+1$ $g(x)=x^2-16$ a) Determine $(f+g)(x)$: $(f+g)(x)=f(x)+g)(x)=3x+1+x^2-16=x^2+3x-15$ b) Determine $(f-g)(x)$: $(f-g)(x)=f(x)-g)(x)=3x+1-(x^2-16)=3x+1-x^2+16=-x^2+3x+17$ c) Determine $(fg)(x)$: $(fg)(x)=f(x)g)(x)=(3x+1)(x^2-16)=3x^3+x^2-48x-16$ d) Determine $\left(\dfrac{f}{g}\right)(x)$: $\left(\dfrac{f}{g}\right)(x)=\dfrac{3x+1}{x^2-16}$ The domain of $\dfrac{f}{g}$ is the set of all real numbers except the zeros of $g$: $x^2-16=0\Rightarrow x^2=16\Rightarrow x=\pm 4$ The domain is: $\left(-\infty,-4\right)\cup\left(-4,4\right)\cup\left(4,\infty\right)$
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