Chapter 13, Problem 6CR

To determine

The graph of the function $f(x)=3^{x-1}+5$ using transformations, and find the domain, range, and horizontal asymptotes of $f$.

Answer to Problem 6CR

Solution:

The domain of $f(x)=3^{x-1}+5$ is $\left(-\infty ,\infty \right)$, range of the function is $\left(5,\infty \right)$, and horizontal asymptote is $y=5$.

The graph of the function is:

Explanation of Solution

Given information:

The function $f(x)=3^{x-1}+5$.

Explanation:

To get the graph of $f(x)=3^{x-1}+5$, begin with the graph of the function $y=3^x$.

For each point $\left(x,3^x\right)$ on the graph of $y=3^x$, the point $\left(x,3^{x-1}\right)$ on the graph of $y=3^{x-1}$ is as indicated in the below table:

$\begin{array}{ccc}x& y=3^x& y=3^{x-1}\\ -1& 0.333& 0.111\\ 0& 1& 0.333\\ 1& 3& 1\\ 2& 9& 3\end{array}$

To obtain the graph of $y=3^{x-1}$, shift horizontally right the graph of $y=3^x$ by $1$ unit.

For each point $\left(x,3^{x-1}\right)$ on the graph of $y=3^{x-1}$, the point $\left(x,3^{x-1}+5\right)$ on the graph of $y=3^{x-1}+5$ is indicated in the below table:

$\begin{array}{ccc}x& y=3^{x-1}& y=3^{x-1}+5\\ -1& 0.111& 5.111\\ 0& 0.333& 5.333\\ 1& 1& 5\\ 2& 3& 8\end{array}$

To obtain the graph of $y=3^{x-1}+5$, shift the graph of $y=3^{x-1}$ vertically upward by $5$ units.

The function $f(x)=3^{x-1}+5$ is an exponential function.

The domain of the exponential function is the set of all real numbers $\left(-\infty ,\infty \right)$.

From the graph, the range of the function $f(x)=3^{x-1}+5$ is the set of all real numbers in the interval $\left(5,\infty \right)$.

The horizontal asymptote is the limit of the function as $x\to \pm \infty$.

Since $x\to -\infty$, $f(x)\to 3^{-\infty -1}+5=0+5=5$.

Hence, horizontal asymptote of $f(x)=3^{x-1}+5$ is $y=5$.

Therefore, the domain of $f(x)=3^{x-1}+5$ is $\left(-\infty ,\infty \right)$, range of the function is $\left(5,\infty \right)$, and horizontal asymptote is $y=5$.