Chapter 11, Problem 9CR

To determine

To graph: Consider the graph of function $f\left(x\right)=3^{x-2}+1$ .

Explanation of Solution

Given information:

The given function is $f\left(x\right)=3^{x-2}+1$ .

Calculation:

Consider the graph of function $f\left(x\right)=3^{x-2}+1$ .

  

The graph is plotted in the complex plane figure-(1) shows.

  

Figure-(1)

Replace $x$ with $x-2$ on the right side $f\left(x\right)=3^x$ .

  $f\left(x\right)=3^{x-2}$

The graph of $f\left(x\right)=3^{x-2}$ is obtain by horizontal shifting the graph $f\left(x\right)=3^x$ to the right by 2 units in figure-(1) shown.

  

Figure-(1)

Add 1 to the right side of $f\left(x\right)=3^{x-2}$ .

  $f\left(x\right)=3^{x-2}+1$

For graphing $f\left(x\right)=3^{x-2}+1$ , we shift the graph of $f\left(x\right)=3^{x-2}$ vertically up by 1unit figure-(1) shown.

  

Figure-(1)

From the graph, we can see that the domain of $f\left(x\right)=3^{x-2}+1$ consists of all real numbers range of $f$ is the set of all numbers $y$ for which $y>1$ and the graph has a horizontal $y=1$ .

Therefore, the domain of $f$ is the set of all real numbers, the range $\left\{y|y>1\right\}$ , and the horizontal asymptote is $y=1$ .