Precalculus (6th Edition)

Published by Pearson
ISBN 10: 013421742X
ISBN 13: 978-0-13421-742-0

Chapter 4 - Inverse, Exponential, and Logarithmic Functions - 4.1 Inverse Functions - 4.1 Exercises: 50

Answer

$f$ and $g$ are inverses of each other.

Work Step by Step

RECALL: (1) $(f \circ g)(x) = f\left(g(x)\right)$ (2) The function $g(x)$ is the inverse of function of a one-to-one function $(x)$ if for every element of the domain, $(f \circ g)(x) =x$ and $(g \circ f)(x)=x$ Find $(f \circ g)(x)$ by substituting $x^2-8$ to $x$ in $f(x)$: $(f\circ g)(x) \\=f\left(g(x)\right) \\=(\sqrt{x^2-8+8}) \\=\sqrt{x^2} \\=|x$ Since $x ]ge 0$, then $|x|=x$ Find $(g \circ f)(x)$ by substituting $\sqrt{x+8}$ to $x$ in $g(x)$: $(g\circ f)(x) \\=g(\left(f(x)\right) \\=(\sqrt{x+8})^2-8 \\=x+8-8 \\=x$ Since $(f\circ g)(x) = (g\circ f)(x)=x$, then $f$ and $g$ are inverses of each other.
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