12. use the graph of the function to find the following: (a) find all x - intercepts of the graph of y =…

Explanation:

Step1: Recall tangent - function properties for (y = \tan x)

x - intercepts

The tangent function (y=\tan x=\frac{\sin x}{\cos x}). The x - intercepts occur when (\sin x = 0) and (\cos x\neq0). So (x = n\pi), where (n\in\mathbb{Z}). In the interval ([- 2\pi,2\pi]), (x=-2\pi,-\pi,0,\pi,2\pi).

Step2: Recall tangent - function properties for (y = \tan x)

y - intercepts

To find the y - intercept, set (x = 0). Then (y=\tan(0)=0).

Step3: Recall tangent - function properties for (y = \tan x)

Increasing and decreasing intervals

The function (y = \tan x) is increasing on the intervals ((-\frac{\pi}{2}+n\pi,\frac{\pi}{2}+n\pi)), (n\in\mathbb{Z}), and it has no decreasing intervals.

Step4: Recall tangent - function properties for (y = \tan x)

Relative extrema

Since (y = \tan x) is always increasing on its domain intervals ((-\frac{\pi}{2}+n\pi,\frac{\pi}{2}+n\pi)), it has no relative extrema.

Step5: Recall tangent - function properties for (y = \tan x)

Vertical asymptotes

The function (y=\tan x) has vertical asymptotes at (x=\frac{\pi}{2}+n\pi), (n\in\mathbb{Z}). In the interval ([-2\pi,2\pi]), (x =-\frac{3\pi}{2},-\frac{\pi}{2},\frac{\pi}{2},\frac{3\pi}{2}).

For (y = \cot x=\frac{\cos x}{\sin x})

Step1: Recall cotangent - function properties for (y=\cot x)

x - intercepts

The x - intercepts occur when (\cos x = 0) and (\sin x\neq0). So (x=\frac{\pi}{2}+n\pi), (n\in\mathbb{Z}). In the interval ([-2\pi,2\pi]), (x =-\frac{3\pi}{2},-\frac{\pi}{2},\frac{\pi}{2},\frac{3\pi}{2}).

Step2: Recall cotangent - function properties for (y=\cot x)

y - intercepts

Set (x = 0), (\cot(0)) is undefined, so there is no y - intercept.

Step3: Recall cotangent - function properties for (y=\cot x)

Increasing and decreasing intervals

The function (y = \cot x) is decreasing on the intervals ((n\pi,(n + 1)\pi)), (n\in\mathbb{Z}), and it has no increasing intervals.

Step4: Recall cotangent - function properties for (y=\cot x)

Relative extrema

Since (y=\cot x) is always decreasing on its domain intervals ((n\pi,(n + 1)\pi)), it has no relative extrema.

Step5: Recall cotangent - function properties for (y=\cot x)

Vertical asymptotes

The function (y=\cot x) has vertical asymptotes at (x=n\pi), (n\in\mathbb{Z}). In the interval ([-2\pi,2\pi]), (x=-2\pi,-\pi,0,\pi,2\pi).

Answer:

For (y = \tan x) (a) (x=-2\pi,-\pi,0,\pi,2\pi) (b) (y = 0) (c) Increasing on ((-\frac{\pi}{2}+n\pi,\frac{\pi}{2}+n\pi),n\in\mathbb{Z}); No decreasing intervals (d) No relative extrema (e) (x=-\frac{3\pi}{2},-\frac{\pi}{2},\frac{\pi}{2},\frac{3\pi}{2})

For (y=\cot x) (a) (x =-\frac{3\pi}{2},-\frac{\pi}{2},\frac{\pi}{2},\frac{3\pi}{2}) (b) No y - intercept (c) Decreasing on ((n\pi,(n + 1)\pi),n\in\mathbb{Z}); No increasing intervals (d) No relative extrema (e) (x=-2\pi,-\pi,0,\pi,2\pi)