Precalculus (6th Edition)

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

Chapter 2 - Graphs and Functions - 2.3 Functions - 2.3 Exercises: 44

Answer

The given relation defines $y$ as a function of $x$. domain: $(-\infty, 0) \cup (0, +\infty)$ range: $(-\infty, 0) \cup (0, +\infty)$

Work Step by Step

Solve for $y$ to obtain: $xy = -6 \\\frac{xy}{x} =\frac{-6}{x} \\y=-\frac{6}{x}$ This means that the given equation is equivalent to $y=-\frac{6}{x}$. The equation above will give only one value of $y$ for every value of $x$. This means that each $x$ is paired with only one value of $y$. Thus, the given relation defines $y$ as a function of $x$. Note that in $y=-\frac{6}{x}$, the value of $x$ cannot be zero because it will make the expression undefined. This means that the domain of the given function is the set of real numbers except $0$. In interval notation, the domain is $(-\infty, 0) \cup (0, +\infty)$. Note that when $-6$ is divided by any non-zero number, the quotient will never be zero. Thus, the value of $y$ can be any real number except zero. In interval notation, the range is $(-\infty, 0) \cup (0, +\infty)$.
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