Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 1 - Precalculus Review - 1.2 Linear and Quadratic Functions - Exercises - Page 19: 55


$f+g$ is a linear function. $f g$ is not linear.

Work Step by Step

Let $f(x)=ax+b, g(x)=cx+d$ be two linear functions. Then, the sum is $(f+g)(x)= (a+c) x + (b+d)$ which is still in a linear form; that is, $f+g$ is a linear function. But for the product $f g$ of $f$ and $g$ is: $(f g)(x) = (ac)(x^2) + (ad+bc) (x) + (b d)$ which is a second-order degree and not linear. Thus, $f g$ is not linear.
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