Chapter 1.3, Problem 4E

To determine

To find: The graph of the function $y=-2^{-x}-1$ . Also, write the domain and range of the function.

Answer to Problem 4E

The graph of the function $y=-2^{-x}-1$ is shown in Figure (1). The domain of the function $y=-2^{-x}-1$ is $\left(-\infty ,\infty \right)$ and the range of the function is $\left(-\infty ,-1\right)$.

Explanation of Solution

Given information:

The function is $y=-2^{-x}-1$.

Calculation:

To graph a function $y=-2^{-x}-1$ , 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 $y=-2^{-x}-1$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{(-)}\boxed{2}\boxed{^}\boxed{(}\boxed{(-)}\boxed{\text{X}}\boxed{)}\boxed{-}\boxed{1}$.

The display will show the equation,

  ${\text{Y}}_{\text{1}}=-2^{-\text{X}}-1$

Now, press the $\boxed{\text{GRAPH}}$ key and $\boxed{\text{TRACE}}$ key to produce the graph of given quadratic equation in standard window as shown below.

  

Figure (1)

From the graph it is observed that the function is defined for all the real values of $x$ . So, the domain of the function $y=-2^{-x}-1$ is the set of real numbers or the interval $\left(-\infty ,\infty \right)$.

From the graph it is observed that the value of $y$ lies in the interval $\left(-\infty ,-1\right)$ for all real values of $x$ . So, the range of the function $y=-2^{-x}-1$ is the interval $\left(-\infty ,-1\right)$.

Therefore, the domain of the function $y=-2^{-x}-1$ is $\left(-\infty ,\infty \right)$ and the range of the function is $\left(-\infty ,-1\right)$.