Precalculus (6th Edition) Blitzer

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

Chapter 7 - Review Exercises - Page 878: 45


The graph is shown below:

Work Step by Step

Let us consider the inequality $ y\le {{2}^{x}}$; substitute the equals symbol in place of the inequality and rewrite the equation as given below: $ y={{2}^{x}}$ To find the value of the x-intercept, substitute y = 0 as given below: $\begin{align} & y={{2}^{x}} \\ & 0={{2}^{x}} \\ & x=\infty \\ \end{align}$ To find the value of the y-intercept, substitute x = 0 as given below: $\begin{align} & y={{2}^{x}} \\ & y={{2}^{0}} \\ & y=1 \\ \end{align}$ We know that it can be concluded from the coordinates that the inequality represents the exponential function which is continuously increasing. Then, take origin $\left( 0,0 \right)$ as a test point and check the region in the graph to shade: $\begin{align} & y\le {{2}^{x}} \\ & 0\overset{?}{\mathop{\le }}\,{{2}^{0}} \\ & 0\le 1 \\ \end{align}$ We see that the above expression is correct, so the shaded region will contain the test point; that is, the region towards the origin must be shaded. Therefore, the curve passes through point $\left( 0,1 \right)$ and never touches the x-axis (that is, the horizontal asymptote is $y=0$). Plot the graph using the intercepts as given below: Thus, the graph of the provided inequality is plotted and the curve passes through $\left( 0,1 \right)$. Also, we get that the shaded region contains the origin.
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