Chapter 4.1, Problem 33E

To determine

To graph: The function $y=5^{-x}+1$ and state the domain, range and asymptote.

Answer to Problem 33E

The graph of the function is shown Figure 1. The domain is $\left(-\infty ,+\infty \right)$ , the range is $\left(1,+\infty \right)$ and the horizontal asymptote is $y=1$ .

Explanation of Solution

Given:

The given function is $y=5^{-x}+1$ and the graph has to be plotted using the graph of $f\left(x\right)=5^x$ .

Calculation:

As observed, the function $y$ can be expressed as $f\left(-x\right)+1$ and so the graph of $y$ is the graph of $f\left(x\right)$ , reflected through the y -axis and then shifted 1 unit upwards, as shown in the image below:

Figure 1

From Figure 1, the domain of the function is $\left(-\infty ,+\infty \right)$ , the range is $\left(1,+\infty \right)$ and the horizontal asymptote is the line $y=1$ .