Chapter 4, Problem 1VE

Solution

To find: The given function and determine its asymptote.

5

Given information: The function $y=-2\cdot {\left(\frac{1}{4}\right)}^{x+1}+5$.

Concept used:

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

Calculation:

To create 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 $x=0$ into $y=-2\cdot {\left(\frac{1}{4}\right)}^{x+1}+5$.

  $\begin{array}{c}y=-2\cdot {\left(\frac{1}{4}\right)}^{0+1}+5\\ y=-2\cdot {\left(\frac{1}{4}\right)}^1+5\\ y=\frac{-2}{4}+5\\ y=\frac{-1}{2}+5\end{array}$

Thus,

  $y=4.5$

Similarly, find the values of $y$ at $x=-2.4,-1.6,0,5,9\text{ and 12}$ and list them in the table given below:

  $\begin{array}{ccccccc}x& -2.4& -1.6& 0& 5& 9& 12\\ y& -8.929& 0.405& 4.5& 4.999& 4.999& 5\end{array}$

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 $y=5$ 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 $\left\{y\in R:y>0\right\}$ . And the line $y=5$ is the asymptote of the given function.