Elementary and Intermediate Algebra: Concepts & Applications (6th Edition)

Published by Pearson
ISBN 10: 0-32184-874-8
ISBN 13: 978-0-32184-874-1

Chapter 12 - Exponential Functions and Logarithmic Functions - 12.1 Composite Functions and Inverse Functions - 12.1 Exercise Set - Page 789: 95

Answer

The pair of provided functions are not inverses of each other.

Work Step by Step

$f\left( x \right)=0.8{{x}^{1/2}}+5.23$and$g\left( x \right)=1.25\left( {{x}^{2}}-5.23 \right),\text{ }x\ge 0$ Now apply the formula of composition of two functions $f\circ g\left( x \right)=f\left( g\left( x \right) \right)$ Substitute $1.25\left( {{x}^{2}}-5.23 \right)$ for $g\left( x \right)$ in the above equation $f\circ g\left( x \right)=f\left( 1.25\left( {{x}^{2}}-5.23 \right) \right)$ Substitute $1.25\left( {{x}^{2}}-5.23 \right)$ for $x$ in the provided function $f\left( x \right)$; the above equation becomes $f\circ g\left( x \right)=0.8{{\left( 1.25\left( {{x}^{2}}-5.23 \right) \right)}^{1/2}}+5.23$ Use the distributive property in the above equation $\begin{align} & f\circ g\left( x \right)=0.8{{\left( 1.25\times {{x}^{2}}-1.25\times 5.23 \right)}^{1/2}}+5.23 \\ & =0.8{{\left( 1.25{{x}^{2}}-6.5375 \right)}^{1/2}}+5.23 \end{align}$ Here, $f\circ g\left( x \right)\ne x$ Here, the composition of two functions is not the identity function.
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