Chapter 3.4, Problem 49E

Solution

To graph: The function $f\left(x\right)=\log \left(x^2\right)$ and analyze it for domain, range, continuity, increasing or decreasing behavior, asymptotes, and end behavior.

Given information:

The function $f\left(x\right)=\log \left(x^2\right)$ .

Graph:

The given function is-

  $f\left(x\right)=\log \left(x^2\right)$

Rewrite the function with the property, $\log \left(a^n\right)=n\log \left(a\right)$ .

  $f\left(x\right)=2\log \left|x\right|$

  

Interpretation:

Domain: $\left(-\infty ,0\right)\cup \left(0,\infty \right)$

Range: $\left(-\infty ,\infty \right)$

Continuity: Function is continuous on its domain.

Increasing or decreasing behavior: Function is decreasing on $\left(-\infty ,0\right)$ and increasing on $\left(0,\infty \right)$ .

Asymptotes: Function has one vertical asymptote at $x=0$ .

End behavior: $\underset{x\to -\infty }{\lim }f\left(x\right)=\infty$ and $\underset{x\to +\infty }{\lim }f\left(x\right)=\infty$ .