College Algebra (6th Edition)

Published by Pearson
ISBN 10: 0-32178-228-3
ISBN 13: 978-0-32178-228-1

Chapter 2 - Functions and Graphs - Exercise Set 2.6 - Page 298: 65

Answer

$a.$ $(f\circ g)(x)=x$ $b.$ $(g\circ f)(x)=x$ $c.$ $(f\circ g)(2)=2$ $d.$ $(g\circ f)(2)=2$

Work Step by Step

$f(x)=\dfrac{1}{x}$ $,$ $g(x)=\dfrac{1}{x}$ $a.$ $(f\circ g)(x)$ To find $(f\circ g)(x)$, substitute $x$ by $g(x)$ in $f(x)$ and simplify: $(f\circ g)(x)=f(g(x))=\dfrac{1}{\Big(\dfrac{1}{x}\Big)}=x$ $b.$ $(g\circ f)(x)$ To find $(g\circ f)(x)$, substitute $x$ by $f(x)$ in $g(x)$ and simplify: $(g\circ f)(x)=g(f(x))=\dfrac{1}{\Big(\dfrac{1}{x}\Big)}=x$ $c.$ $(f\circ g)(2)$ Substitute $x$ by $2$ in $(f\circ g)(x)$, which was found in part $a$, and evaluate: $(f\circ g)(2)=f(g(2))=2$ $d.$ $(g\circ f)(2)$ Substitute $x$ by $2$ in $(g\circ f)(x)$, which was found in part $b$, and evaluate: $(g\circ f)(2)=g(f(2))=2$
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