College Algebra 7th Edition

Published by Brooks Cole
ISBN 10: 1305115546
ISBN 13: 978-1-30511-554-5

Chapter 2, Functions - Section 2.3 - Getting Information from the Graph of a Function - 2.3 Exercises: 25

Answer

(a) $x=\left\{-2, 1\right\}$ (b) $[-2, 1]$

Work Step by Step

To solve the given equation graphically, perform the following steps: (1) Treat each side of the equation as a function and graph them on the same coordinate plane. Use a graphing calculator to graph: $y=x^2$ (red graph) $y=2-x$ (blue graph) (refer to the image below for the graph) (2) Identify the point/s where the graphs intersect (a) The points where the two graphs intersect represents the instance where the value of $x^2$ is equal to the value of $2-x$. The x-coordinate of this point is the solution to the given equation. Notice that the graphs intersect at the points $(-2, 4)$ and $(1, 1)$. The x-coordinates of these points are $-2$ and $1$. Thus, the solution to the given equation is $x=\left\{-2, 1\right\}$. (b) The region where the graph of the function $y=x^2$ is lower than or equal to the graph of the function $y=2-x$ is the region where $x^2 \le 2-x$. Notice that the red graph ($y=x^2$) is lower than or equal to the blue graph ($y=2-x$) in the region $(-2, 1)$. Since the inequality involves $\le$, then the endpoints $-2$ and $1$ are part of the solution set. Thus, the solution to the given inequality is $[-2, 1]$.
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