Chapter 1, Problem 29RE

To determine

To find: The graph for the member of the family of given functions for several values of $c$ and then determine how the graph changes.

Answer to Problem 29RE

The required graph is shown in Figure 1 and the graph is deeper at $x=0$ as the value of $c$ increases.

Explanation of Solution

Given:

The given function is $f\left(x\right)=\ln \left(x^2-c\right)$ .

Calculation:

Consider the given functions equations,

  $f\left(x\right)=\ln \left(x^2-c\right)$

The graph for the given function for the different values of $c$ is shown in Figure 1

  

Figure 1

The graph shows it is seen that for $c<0$ the function is defined at every point and as the value of $c$ increases the dip of the graph at $x=0$ increases.

Consider the graph for $c\ge 0$ .

  $\underset{x\to \pm \sqrt{c}}{\text{lim}}\ln \left(x^2-c\right)=-\infty$

Thus, the graph asymptotes at $x=\sqrt{c}$ .