Chapter 9.7, Problem 25SR

To determine

To plot the graph $y=3^x$ , find the y-intercept and state the domain and range.

Answer to Problem 25SR

  $\begin{array}{l}y-\text{intercept:}\left(0,1\right)\\ \text{Domain:}\left(-\infty ,\infty \right)\\ \text{Range}:\left(0,\infty \right)\end{array}$

Explanation of Solution

Given information:

  $y=3^x$

Concept used:

• Domain of a function $y=f\left(x\right)$ is the set of x values for which the function is defined.
• Range of a function $y=f\left(x\right)$ is the set of y -values that the function can take.
• x -intercept is the point where the graph of the function crosses/touches the x -axis.
• y -intercept is the point where the graph of the function crosses/touches the y -axis.

Calculation:

In order to get the graph of $f\left(x\right)=3^x$ , first make a table by taking some random values of x and find the corresponding values of y . And then plot the ordered pairs and connect the points with a smooth curve, so the table is:

X	$f\left(x\right)=3^x$	Y
- 2	$f\left(x\right)=3^x$	.11
- 1	$f\left(x\right)=3^x$	.33
0	$f\left(x\right)=3^x$	1
1	$f\left(x\right)=3^x$	3
2	$f\left(x\right)=3^x$	9

Graph of the function is shown below:

  

From the above graph it is clear that the curve intersecting the y -axis at the point (0, 1). So, the y -intercept of the function is at $y=1$ .

Since the graph is moving form left to right without any discontinuity, so the function is defined for all values of x . Thus, the domain of the function is all real numbers, that is, $\text{Domain}=\left(-\infty ,\infty \right)$ .

Also, from the graph it is clear that function can take only the y -values that are above the x -axis. So, the range of the function will be all real number greater than 0, that is, $\text{Range}=\left(0,\infty \right)$