Chapter 7.5, Problem 22E

To determine

The graph of the inequality $y\ge 3^{-x}-2$ using graphing utility

Answer to Problem 22E

The point $\left(1,0\right)$ satisfies inequality $y=3^{-x}-2$ shading that region which contains point $\left(0,1\right)$

Explanation of Solution

Given:

The graph of the inequality $y\ge 3^{-x}-2$

Formula used:

The substitution method is used.

Calculation:

To sketch the graph of the given inequality, $y\ge 3^{-x}-2$

Consider given inequality with equality sign.

  $y=3^{-x}-2$

Graph the equation $y=3^{-x}-2$

Since inequality is of >, the graph with solid line is drawn

Now consider point $\left(0,1\right)$

Evaluate inequality $y=3^{-x}-2$ for point $\left(0,1\right)$

  $\begin{array}{l}1\ge 3^{-0}-2\\ \Rightarrow 1\ge 1-2\\ \Rightarrow 1\ge -1\end{array}$

As point $\left(1,0\right)$ satisfies inequality $y=3^{-x}-2$ shading that region which contains point $\left(0,1\right)$

The Graph of given inequality $y=3^{-x}-2$ is shown below:

  

Conclusion:

The point $\left(1,0\right)$ satisfies inequality $y=3^{-x}-2$ shading that region which contains point $\left(0,1\right)$