Chapter 11, Problem 9CR

To determine

To graph: The function $f(x)=3^{x-2}+1$ using transformations. Find the domain, range, and horizontal asymptotes of $f$.

Explanation of Solution

Given information:

The function $f(x)=3^{x-2}+1$.

Graph:

To get the graph of $f(x)=3^{x-2}+1$ 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 $\left(x,3^{x-2}\right)$ on the graph of $y=3^{x-2}$ as indicated below table,

$\begin{array}{ccc}x& y=3^x& y=3^{x-2}\\ -1& 0.333& 0.037\\ 0& 1& 0.111\\ 1& 3& 0.333\\ 2& 9& 1\end{array}$

To obtain the graph of $y=3^{x-2}$, shift horizontally right the graph of $y=3^x$ by $2$ unit.

For each point $\left(x,3^{x-2}\right)$ on the graph of $y=3^{x-2}$, the point $\left(x,3^{x-2}+1\right)$ on the graph of $y=3^{x-2}+1$ as indicated below table,

$\begin{array}{ccc}x& y=3^{x-2}& y=3^{x-2}+1\\ -1& 0.037& 1.037\\ 0& 0.111& 1.111\\ 1& 0.333& 1.333\\ 2& 1& 2\end{array}$

To obtain the graph of $y=3^{x-2}+1$, shift upwardvertically the graph of $y=3^{x-2}$ by $1$ unit.

The function $f(x)=3^{x-2}+1$ is exponential function.

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

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

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

Since as $x\to -\infty$, $f(x)\to 3^{-\infty -2}+1=0+1=1$.

Hence, horizontal asymptote of $f(x)=3^{x-2}+1$ is $y=1$.

Therefore, the domain of $f(x)=3^{x-2}+1$ is $\left(-\infty ,\infty \right)$, range of the function is $\left(1,\infty \right)$ and horizontal asymptote is $y=1$.

Interpretation:

To graph the function $f(x)=3^{x-2}+1$, use the graph of $y=3^x$, shift the graph horizontally by $2$ units to the right, then shift the graph vertically upward by $1$ unit.

The domain of $f(x)=3^{x-2}+1$ is $\left(-\infty ,\infty \right)$, range of the function is $\left(1,\infty \right)$ and horizontal asymptote is $y=1$.