Chapter 4, Problem 8REAE

Solution

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

  $\left\{x:x\in ℝ\right\}$ , $\left\{y:y\le -2,y\in ℝ\right\}$

Given:

A function $f\left(x\right)=-3\cdot {\left(4\right)}^{x+1}-2$ is given.

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 $y$ as follows:

Substitute $x=0$ in $f\left(x\right)=-3\cdot {\left(4\right)}^{x+1}-2$ :

  $\begin{array}{c}y=-3{\left(4\right)}^{x+1}-2\\ y=-14\end{array}$

Similarly, find the values of $y$ at $x=-1,0,1$ and list them in the table given below:

  $\begin{array}{cccc}x& -1& 0& 1\\ y& -5& -14& -50\end{array}$

Now, graph of the function as follows;

  

From the observation of the graph of the function $y=3{\left(2\cdot 5\right)}^x$ it is clear that at $x=-6$ the value of $y$ is $-2$ therefore asymptote is $y=-2$ hence the range is $y\le -2$

Here function is well defined for all value of $x$ .

Conclusion:

Hence, domain of the function $y=3{\left(2\cdot 5\right)}^x$ is $x\in ℝ$ while range is $y\le -2$ .