Answer:
x-intercept of the graph of the function
is, ![(\frac{\pi}{6} \pm \frac{n\pi}{3} , 0)](https://tex.z-dn.net/?f=%28%5Cfrac%7B%5Cpi%7D%7B6%7D%20%5Cpm%20%5Cfrac%7Bn%5Cpi%7D%7B3%7D%20%2C%200%29)
Step-by-step explanation:
Given the function:
......[1]
x-intercept defined as the graph crosses the x-axis.
Substitute value of y = 0 in [1] and solve for x;
or
Take the inverse cotangent of both sides of the equation and solve for x;
3x = arccot (0)
We know the exact value of ![arc\cot(0) = \frac{\pi}{2}](https://tex.z-dn.net/?f=arc%5Ccot%280%29%20%3D%20%5Cfrac%7B%5Cpi%7D%7B2%7D)
then;
![3x = \frac{\pi}{2}](https://tex.z-dn.net/?f=3x%20%3D%20%5Cfrac%7B%5Cpi%7D%7B2%7D)
Divide both sides by 3 we get;
![x = \frac{\pi}{6}](https://tex.z-dn.net/?f=x%20%3D%20%5Cfrac%7B%5Cpi%7D%7B6%7D)
Since, the cotangent function is positive in the first and third quadrants.
The period of the function
is
so values will repeat every
radians in both directions.
we have;
![x =\frac{\pi}{6} \pm \frac{n\pi}{3}](https://tex.z-dn.net/?f=x%20%3D%5Cfrac%7B%5Cpi%7D%7B6%7D%20%5Cpm%20%5Cfrac%7Bn%5Cpi%7D%7B3%7D)
Therefore, the x-intercept of the graph of the function
is ;
for every integer n;