Precalculus (6th Edition) Blitzer

Published by Pearson
ISBN 10: 0-13446-914-3
ISBN 13: 978-0-13446-914-0

Chapter 2 - Mid-Chapter Check Point - Page 381: 9

Answer

Domain of $f\left( x \right):\left( -\infty ,\infty \right)$ , Range of $f\left( x \right):\left( -\infty ,5 \right]$
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Work Step by Step

The graph is a parabola. Also, since $f\left( x \right)$ is a polynomial, it exists for every value of $x$ on the real axis. Therefore, the domain of $f\left( x \right)=5-{{\left( x+2 \right)}^{2}}$ is $\left( -\infty ,\infty \right)$. And, since, $\begin{align} & {{\left( x+2 \right)}^{2}}\ge 0 \\ & -{{\left( x+2 \right)}^{2}}\le 0 \\ & 5-{{\left( x+2 \right)}^{2}}\le 5 \\ & f\left( x \right)\le 5 \\ \end{align}$ $\Rightarrow f\left( x \right)\in \left( -\infty ,5 \right]$ So, the range of the function is $\left( -\infty ,5 \right]$.
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