Calculus 10th Edition

Published by Brooks Cole
ISBN 10: 1-28505-709-0
ISBN 13: 978-1-28505-709-5

Chapter P - P.1 - Graphs and Models - Exercises - Page 8: 26

Answer

$y$-intercepts: (0, 1) and (0, -1) $x$-intercepts: $\left(\frac{\sqrt 3}{3}, 0\right)$ and $\left(-\frac{\sqrt 3}{3}, 0\right)$

Work Step by Step

To find the $y$-intercept(s), set $x=0$. \begin{align*} y&=2(0)-\sqrt(0^2+1) \\ &=0-\sqrt (1) \\ &=0-(\pm 1) \\ &=\pm 1 \end{align*} Thus, the $y$-intercepts are (0, 1) and (0, -1). To find the $x$-intercept(s), set $y=0$. \begin{align*} 0&=2x-\sqrt(x^2+1) \\ \sqrt(x^2+1)&=2x \\ x^2+1&=4x^2 \\ 1&=3x^2 \\ \frac{1}{3}&=x^2 \\ x&=\sqrt\frac{1}{3} \\ &=\frac{\sqrt 1}{\sqrt 3} \\ &=\frac{\pm 1}{\sqrt 3} \\ &=\frac{\pm 1}{\sqrt 3}\left(\frac{\sqrt 3}{\sqrt 3}\right) \\ &=\pm\frac{\sqrt 3}{3} \end{align*} Thus, the $x$-intercepts are $\left(\frac{\sqrt 3}{3}, 0\right)$ and $\left(-\frac{\sqrt 3}{3}, 0\right)$.
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