Chapter 4.6, Problem 17E

Solution

To graph : The function $y=-{\tan }^{2}x$ showing four periods and its domain and range.

Given information :

The function $y=-{\tan }^{2}x$

Graph :

Domain:

The domain of $f\left(x\right)$ is the set of all those real numbers for which $f\left(x\right)$ is meaningful.

Range:

Range of a real number of a real variable is the set of all real values taken by $f\left(x\right)$ at points in its domain.

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

Graph the above equation in the viewing window $\left[-2\pi ,2\pi \right]\text{ by}\left[-30,8\right]$ .

  

Hence, it is evident from the graph that for all the values of $x$ , except the odd integral multiples of $\frac{\pi }{2}$ ,the function is defined. Thus, the domain of the function is $x\in ℝ-\left(2n+1\right)\frac{\pi }{2}$ .

And, from the graph it is observed that the function assumes all real values between negative infinity and $0$ . Thus, the range of the function is $y\in \left(-\infty ,0\right)$ .

Interpretation :

The domain of the function is $x\in ℝ-\left(2n+1\right)\frac{\pi }{2}$ .

The range of the function is $y\in \left(-\infty ,0\right)$ .