The x-intercept of the graph of the function y=tan(x-5π/6) is correctly given by: Option D: (5π/6, 0)
The x-intercept of a function of variable x ( y = f(x) ) form is an intersection fo the x-axis and the curve of the function.
The x-intercept for a function y = f(x) is a solution to the equation f(x) = 0 because at that value of x, the function f(x) lies on x-axis, where y is 0. Values of x-intercept for a function f(x) are also called roots or solution of f(x) = 0 equation.
For this case, the function considered is:
[tex]y = \tan\left( x - \dfrac{5\pi}{6} \right)[/tex]
For its x-intercept, putting y = 0, we get:
[tex]0 = \tan\left( x - \dfrac{5\pi}{6} \right)\\\\ \tan\left( x - \dfrac{5\pi}{6} \right) = 0\\\\[/tex]
tan is 0 when the input of tan is [tex]\pi n[/tex]radians, for any integral value of n
Thus, we get:
[tex]x - \dfrac{5\pi}{6} = \pi n[/tex]
Or, we get:
[tex]x = \pi n + \dfrac{5\pi}{6} = \dfrac{\pi (6n + 5)}{6} \: \rm \forall \: n \in \mathbb Z[/tex]
Thus, the coordinates of x-intercepts of the considered function are of the form:
[tex](x, y) = (x, 0) = \left( \dfrac{\pi (6n + 5)}{6} , 0\right)[/tex]
Checking all the options:
If that's true, then we get:
[tex]\dfrac{\pi (6n + 5)}{6} = -\dfrac{2\pi}{3}\\\\6n + 5 = -4\\\\n = \dfrac{-9}{6} \neq \: \rm integer[/tex]
Thus, this option is incorrect.
If that's true, then we get:
[tex]\dfrac{\pi (6n + 5)}{6} = -\dfrac{\pi}{3}\\\\6n + 5 = -2\\\\n = \dfrac{-7}{6} \neq \: \rm integer[/tex]
Thus, this option is incorrect.
If that's true, then we get:
[tex]\dfrac{\pi (6n + 5)}{6} = \dfrac{\pi}{6}\\\\6n + 5 = 1\\\\n = \dfrac{-4}{6} \neq \: \rm integer[/tex]
Thus, this option is incorrect.
If that's true, then we get:
[tex]\dfrac{\pi (6n + 5)}{6} = \dfrac{5\pi}{6}\\\\6n + 5 = 5\\\\n = \dfrac{0}{6} = 0[/tex]and this time, n is integer.
Thus, this option is correct.
Thus, the x-intercept of the graph of the function y=tan(x-5π/6) is correctly given by: Option D: (5π/6, 0)
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