Chapter 0.3, Problem 15E

To determine

To Match: The given function with its given graph.

Answer to Problem 15E

The graph of the function $y=-3^{-x}$ is option $\left(e\right)$ .

Explanation of Solution

Given:

  $y=-3^{-x}$

Options:

(a)

(b)

(c)

(d)

(e)

(f)

Calculation:

Find the graph of $y=-3^{-x}$

The $y-$ intercept of this function is:

  $\begin{array}{l}y=-3^{-0}\\ y=-1\end{array}$

Which narrows down our options to $\left(b\right),\left(c\right),\left(e\right),$ or $\left(f\right)$ . Note that the function increases as $x$ increases:

  $\begin{array}{l}-3^{-1}=-\frac{1}{3}\\ -3^{-2}=-\frac{1}{9}\\ -3^{-3}=-\frac{1}{27}\end{array}$

which narrows down our options to $\left(e\right)$ or $\left(f\right)$ . Note that $\left(f\right)$ has an $x$ -intercept.

So, the correct match for the function $y=-3^{-x}$ is option $\left(e\right)$ .

Find an $x-$ intercept for this function:

  $\begin{array}{c}-3^{-x}=0\\ 3^{-x}=0\end{array}$

Taking the logarithm of both sides of the equation is the only way to solve this equation.

The logarithm of $0$ cannot be taken. As a result, the function does not have an $x-$ intercept, and $\left(e\right)$ is the correct answer.