Chapter 6.3, Problem 68E

a.

To determine

To compare the end behavior of the logarithmic function $g$ to that of the exponential function $f$

Answer to Problem 68E

The behavior of the logarithmic function $g$ to that of the exponential function $f$

  $f\left(x\right)=e^x\text{ and }g\left(x\right)=\ln \left(x\right)$

Explanation of Solution

Given information:

Consider the given information,

  $f\left(x\right)$ Is exponential function and $g\left(x\right)$ is logarithmic function.

As from the information, it is clear that $f\left(x\right)=e^x$ which is an exponential function and $g\left(x\right)=\ln x$ which is a logarithmic function.

b.

To determine

To determine whether the function are inverse function.

Answer to Problem 68E

The inverse function is $f^{-1}\left(x\right)=g\left(x\right)$ for $f$ and for $g$ is $g^{-1}\left(x\right)=f\left(x\right)$

Explanation of Solution

Given information:

Consider the information given,

  $f\left(x\right)$ Is an exponential function and $g\left(x\right)$ is a logarithmic function.

Calculation:

The inverse function is given,

  $\begin{array}{l}f\left(x\right)=e^x=y\\ =x=\ln y\\ =f^{-1}\left(x\right)=\ln x=g\left(x\right)\\ =f^{-1}\left(x\right)=g\left(x\right)\\ \text{similarly }{g}^{-1}\left(x\right)=f\left(x\right)\end{array}$

The inverse function of $f$ is $=f^{-1}\left(x\right)=g\left(x\right)$ and inverse function of $g$

  $=g^{-1}\left(x\right)=f\left(x\right)$

c.

To determine

To determine the base of each function

Answer to Problem 68E

The base of both function are $e$ .

Explanation of Solution

Given information:

Consider the information given,

  $f\left(x\right)$ Is an exponential function and $g\left(x\right)$ is a logarithmic function.

Both the functions are inverse of each other.

Base of both functions are $e$ .