Chapter 4.3, Problem 49E

Solution

To graph: The given function and determine the domain and range.

Domain $=\left(-\infty ,\infty \right)$

Range: $y>1$

Given:

A function $g\left(x\right)=y=\frac{4}{3}{e}^{x-1}+1$ is given.

Concept used:

1. A exponential function having form $y=a{e}^{rx}$ is said to growth function if $a>0$ and $r>0$ and it is said to be decay function if $a>0$ and $r<0$ .
2. The domain is a set of all real numbers for any natural base functions.
3. The range is the set of real numbers above or below the horizontal asymptote $y=h$ . It means if $a>0$ then $f\left(x\right)>h$ and if $a<0$ then $f\left(x\right)<h$ .

Calculation:

The comparison of function $y=\frac{4}{3}{e}^{x-1}$ with $y=a{e}^{rx}$ gives $a=\frac{4}{3}$ and $r=1$ so the function is exponential growth function because $a>0$ and $r>0$ .

Evaluate the function at different values of $x$ .

When $x=0$ the value of $y$ is

  $\begin{array}{l}y=\frac{4}{3}{e}^{0-1}+1\\ y=0.490+1\\ y=1.49\end{array}$

When $x=1$ the value of $y$ is

  $\begin{array}{l}y=\frac{4}{3}{e}^{1-1}+1\\ y=\frac{4}{3}+1\\ y=2.33\end{array}$

To sketch the graph, plot the points $\left(0,1.49\right)$ and $\left(1,2.33\right)$ and draw the curve.

  

The domain of the function is set of all real numbers. So,

Domain $=\left(-\infty ,\infty \right)$

It is clear from the graph that the horizontal asymptote is $y=1$ . So, the range is,

  $y>1$

Conclusion:

The domain and range of function is $\left(-\infty ,\infty \right)$ and $y>1$ respectively.