Calculus 10th Edition

Published by Brooks Cole
ISBN 10: 1-28505-709-0
ISBN 13: 978-1-28505-709-5

Chapter 5 - Logarithmic, Exponential, and Other Transcendental Functions - 5.3 Exercises: 6

Answer

a) The functions are inverses of each other. b) Please see "step by step"

Work Step by Step

If f(x) and g(x) are inverses (of each other, then a. $f(g(x))=x$ and $g(f(x))=x$ for x in respective domains (of g and f). (definition, p.337) b. The graphs of f and g are reflections of each other across the line x=y. (Theorem 5.6, figure 5.12) ------------------ a. $f(g(x)=16-[g(x)]^{2}=16-(\sqrt{16-x})^{2}$ $=16-(16-x)=x$ $g(f(x)) = \sqrt{16-f(x)}=\sqrt{16-(16-x^{2})}$ $=\sqrt{x^{2}}=|x|$ This equals $x$ because the domain of $g(f(x))$ is the domain of $f(x),$ which is $x\geq 0$, for which $|x|=x.$ b. Using a graphing utility, (see below) we see that the blue graph (f(x) ) and the red graph (g(x)) are reflections of each other about the line x=y: .
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