Chapter 9.6, Problem 15PPS

To determine

To plot the graph $y=3\cdot {11}^x$ , find the value of $3{\left(11\right)}^{-0.2}$ , find the y-intercept and state the domain and range.

Answer to Problem 15PPS

  $\begin{array}{l}y-\text{intercept:}\left(0,3\right)\\ \text{Domain:}\left(-\infty ,\infty \right)\\ \text{Range}:\left(0,\infty \right)\\ 3{\left(11\right)}^{-0.2}=1.857\end{array}$

Explanation of Solution

Given information:

  $y=3\cdot {11}^x$

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 $y=3\cdot {11}^x$ , 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	$y=3\cdot {11}^x$	y
- 2	$y=3\cdot {11}^{-2}$	0.02
- 1	$y=3\cdot {11}^{-1}$	0.27
0	$y=3\cdot {11}^0$	3
1	$y=3\cdot {11}^1$	33
2	$y=3\cdot {11}^2$	363

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, 3). So, the y -intercept of the function is at $y=3$ .

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, the range of the function will be all real number greater than 0, that is, $\text{Range}=\left(0,\infty \right)$ .

And the value of $3{\left(11\right)}^{-0.2}$ is 1.857 from the above graph.