Chapter 4, Problem 6T

Solution

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

  $\text{Domain}=\left\{x:x\in R\right\}\text{ and Range}=\left\{y\in R:y>0\right\}$

Given information:

A function $g\left(x\right)={\left(\frac{2}{3}\right)}^x+2$.

Concept used:

1. The graph of a function can be drawn by finding the points that lies on the curve of the given function.
2. The domain of an exponential function is the set of all real numbers.

Calculation:

To make the graph of the given function, substitute different values of $x$ in the given function and find the respective value of $g\left(x\right)$ as follows:

Substitute $x=0$ into $g\left(x\right)={\left(\frac{2}{3}\right)}^x+2$ :

  $\begin{array}{c}g\left(0\right)={\left(\frac{2}{3}\right)}^0+2\\ =2\left[a^0=1\right]\end{array}$

Similarly, find the values of $g\left(x\right)$ at $x=-6,-4,0,5\text{ and 10}$ and list them in the table given below.

  $\begin{array}{cccccc}x& -6& -4& 0& 5& 10\\ y& 13.39& 7.06& 3& 2.13& 2.01\end{array}$

Now, draw each point of form $\left(x,y\right)$ in a rectangular coordinate system and connect them with a smooth curve as follows.

  

From the graph, it is clear that the line $y=0$ is the asymptote of the given function; therefore, the domain of the given function is the entire set of real numbers and the range of the given function is set of all positive real numbers.

Conclusion:

Hence, the domain of the given function is the set $\left\{x:x\in R\right\}$ , and the range of the given function is the set $\left\{y\in R:y>0\right\}$.