Chapter 1, Problem 31RE

(a)

To determine

To find: The domain of the function $y=2e^{-x}-3$ .

Answer to Problem 31RE

The domain of the function $y=2e^{-x}-3$ is the interval $\left(-\infty ,\infty \right)$ .

Explanation of Solution

Given information:

The given function is $y=2e^{-x}-3$ .

Calculation:

The function $y=2e^{-x}-3$ is an exponential function and it is defined for all the real values of $x$ .

Therefore, the domain of the function $y=2e^{-x}-3$ is the set of real numbers or the interval $\left(-\infty ,\infty \right)$ .

(b)

To determine

To find: The range of the function $y=2e^{-x}-3$ .

Answer to Problem 31RE

The range of the function $y=2e^{-x}-3$ is the interval $\left(-3,\infty \right)$ .

Explanation of Solution

Given information:

The given function is $y=2e^{-x}-3$ .

Calculation:

The value of the function $y=2e^{-x}-3$ is an exponential function that is always greater than $-3$ .

Therefore, the range of the function $y=2e^{-x}-3$ is the interval $\left(-3,\infty \right)$ .

(c)

To determine

To graph: The function $y=2e^{-x}-3$ .

Explanation of Solution

Given information:

The given functionis $y=2e^{-x}-3$ .

Graph:

To graph a function $y=2e^{-x}-3$ , follow the steps using graphing calculator.

First press “ON” button on graphical calculator, press $\boxed{\text{Y}=}$ key and enter right hand side of the equation $y=2e^{-x}-3$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{2}\boxed{\text{2nd}}\boxed{\text{LN}}\boxed{(-)}\boxed{\text{X}}\boxed{)}\boxed{-}\boxed{3}$ .

The display will show the equation,

  ${\text{Y}}_{\text{1}}=2e^\left(-\text{X}\right)-3$

Now, press the $\boxed{\text{GRAPH}}$ key and $\boxed{\text{TRACE}}$ key to produce the graph of both the equations as shown below.

  

Figure (1)

Interpretation: From the above graph it can be observed that the graph of the function is $y=2e^{-x}-3$ formed by shifting the graph of its parent function $f\left(x\right)=e^{-x}$ by 2 units downward and stretches toward x -axis.