Chapter 1.3, Problem 4QQ

(a)

To determine

To find:The domain of the function $f\left(x\right)=e^{-x}-2$ .

Answer to Problem 4QQ

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

Explanation of Solution

Given information:The function is $f\left(x\right)=e^{-x}-2$ .

Calculation:

Consider the given function.

  $f\left(x\right)=e^{-x}-2$

The function is defined for all real values of $x$ .So, the domain of the function is the set of real numbers or the interval $\left(-\infty ,\infty \right)$ .

Therefore, the domain of the function $f\left(x\right)=e^{-x}-2$ is $\left(-\infty ,\infty \right)$ .

(b)

To determine

To find:The range of the function $f\left(x\right)=e^{-x}-2$ .

Answer to Problem 4QQ

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

Explanation of Solution

Given information:The function is $f\left(x\right)=e^{-x}-2$ .

Calculation:

The range of an exponential function of the form $c\cdot e^{ax+b}+k$ is $f\left(x\right)>k$ . Consider the given function.

  $f\left(x\right)=e^{-x}-2$

So, for the given function $f\left(x\right)>-2$ . The range is $\left(-2,\infty \right)$ .

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

(c)

To determine

To find:The zeros of the function $f\left(x\right)=e^{-x}-2$ .

Answer to Problem 4QQ

The zeros of the function $f\left(x\right)=e^{-x}-2$ is $x=-0.693$ .

Explanation of Solution

Given information:The function is $f\left(x\right)=e^{-x}-2$ .

Calculation:

To get the zeros of the function, equate the function to zero.

  $\begin{array}{c}f\left(x\right)=0\\ e^{-x}-2=0\\ e^{-x}=2\end{array}$

Take logarithm on both the sides of above equation.

  $\begin{array}{c}\ln \left(e^{-x}\right)=\ln 2\\ -x=\ln 2\\ x=-0.693\end{array}$

Therefore, the zeros of the function $f\left(x\right)=e^{-x}-2$ is $x=-0.693$ .