Intermediate Algebra (12th Edition)

Published by Pearson
ISBN 10: 0321969359
ISBN 13: 978-0-32196-935-4

Chapter 2 - Section 2.5 - Introduction to Relations and Functions - 2.5 Exercises: 51

Answer

$y \text{ is NOT a function of }x \\\text{Domain: } [0,\infty)$

Work Step by Step

$\bf{\text{Solution Outline:}}$ To determine if the given equation, $ x=y^6 ,$ is a function, solve first for $y$ in terms of $x.$ Then check if $x$ is unique for every value of $y.$ To find the domain, find the set of all possible values of $x.$ $\bf{\text{Solution Details:}}$ Using the properties of equality, the given equation is equivalent to \begin{array}{l}\require{cancel} x=y^6 \\\\ y^6=x \\\\ y=\pm\sqrt[6]{x} .\end{array} If $x=64,$ then $y=2$ or $y=-2.$ That is, the ordered pairs, $\{ (64,2),(64,-2) \}$ satisfy the given equation. This means that the value of $x$ is not unique. Hence, $y$ is NOT a function of $x.$ Since $x$ appears in the radicand of a radical with an even index, then \begin{array}{l}\require{cancel} x\ge0 .\end{array} Hence, the domain is the set of all nonnegative numbers. Hence, the given equation has the following characteristics: \begin{array}{l}\require{cancel} y \text{ is NOT a function of }x \\\text{Domain: } [0,\infty) .\end{array}
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