Chapter 3.1, Problem 49E

a.

To determine

Draw the graph of the function .

Explanation of Solution

Given:

It is given in the question that the function is $f(x)=\frac{8}{1+e^{-0.5x}}$ .

Graph:

  

Interpretation: In this,the red colour represents the graph of the function $f(x)=\frac{8}{1+e^{-0.5x}}$ .

In the graph , just put the function on it,it gives the required graph.

b.

To determine

Find the asymptotes numerically from the function .

Answer to Problem 49E

The asymptotes for the function are $y=0,y=8$ .

Explanation of Solution

Given:

It is given in the question that the function is $f(x)=\frac{8}{1+e^{-0.5x}}$ .

Concept Used:

In this,use the concept of asymptotes,there are two types of asymptotes:vertical asymptotes and horizontal asymptotes.

Calculation:

Horizontal asymptotes: $y=c$ is the horizontal asymptotes for $x\to \infty$ .

As seen from the graph, two horizontal asymptotes is

for $x\to +\infty ,y=\frac{8}{1+e^{-0.5(\infty )}}\Rightarrow y=\frac{8}{1+e^{-\infty }}\Rightarrow y=\frac{8}{1+\frac{1}{\infty }}=8$

  $x\to -\infty ,y=\frac{8}{1+e^{-0.5(-\infty )}}\Rightarrow y=\frac{8}{1+e^{\infty }}\Rightarrow y=\frac{8}{1+\infty }=0$

So , the two asymptotes are $y=0,y=8$ .

Conclusion:

The two asymptotes are $y=0,y=8$ .