Chapter 11.3, Problem 16E

(a)

To determine

To find: The domain of the function.

Answer to Problem 16E

The domain is all the real numbers $e^x$ that is defined for all the real numbers.

Explanation of Solution

Given:

The given function is $f\left(x\right)=\frac{e^x}{e^x+c}$ . For C to be greater than the 0.

Calculation:

Consider the given function,

  $f\left(x\right)=\frac{e^x}{e^x+c}$

The above function is defined as $e^x+c\ne 0$

Since, $c>0$ and $e^x>0$ it is concluded that the $e^x+c>0$ , that is always different other than 0.

Thus, the domain is all the real numbers $e^x$ that is defined for all the real numbers.

(b)

To determine

To find: The range of the function.

Answer to Problem 16E

The range is (0, 1).

Explanation of Solution

Consider the given function,

  $f\left(x\right)=\frac{e^x}{e^x+c}$

Then,

  $\begin{array}{c}f\left(x\right)=\frac{e^x+c-c}{e^x+c}\\ =\frac{e^x+c}{e^x+c}-\frac{c}{e^x+c}\end{array}$

In the above equation $1-\frac{c}{e^x+c}<1$ as the value of $\frac{c}{e^x+c}>0$ .

Then, it is concluded that $0<f\left(x\right)<1$ and $\underset{x\to \infty }{\lim }f\left(x\right)=1$ also, $\underset{x\to -\infty }{\lim }f\left(x\right)=0$

Thus, the range is (0, 1).

(c)

To determine

To find: The way in which the value of the c affect the graph.

Answer to Problem 16E

The graph is shifted to the right for $c>1$ and to the left for $c<1$ .

Explanation of Solution

Consider that graphing calculator use will show that for the value of $c$ the graph is shifted to the right for $c>1$ and to the left for $c<1$ .