Calculus (3rd Edition)

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

Chapter 1 - Precalculus Review - 1.3 The Basic Classes of Functions - Exercises - Page 23: 43

Answer

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Work Step by Step

Compute $\delta (f+g)(x)$: $\delta (f+g)(x)=\delta (f(x)+g(x))=(f(x+1)+g(x+1))-(f(x)+g(x))$ $=f(x+1)+g(x+1)-f(x)-g(x)$ $=(f(x+1)-f(x))+(g(x+1)-g(x))$ $=\delta f(x)+\delta g(x)$ We got: $\delta (f+g)(x)=\delta f(x)+\delta g(x)$ $\delta (f+g)=\delta f+\delta g$ Compute $\delta (cf)(x)$: $\delta (cf)(x)=(cf)(x+1)-(cf)(x)$ $=cf(x+1)-cf(x)$ $=c(f(x+1)-f(x))$ $=c\delta f(x)$ We got: $\delta (cf)(x)=c\delta f(x)$ $\delta (cf)=c\delta (f)$
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