To determine which point the graph of the parent function [tex]\( y = \tan(x) \)[/tex] passes through, we need to evaluate the tangent function for specific values of [tex]\( x \)[/tex] and compare those results to the given points.
1. Consider the value [tex]\( x = \frac{\pi}{3} \)[/tex].
- Calculate [tex]\( y = \tan\left(\frac{\pi}{3}\right) \)[/tex].
The tangent of [tex]\( \frac{\pi}{3} \)[/tex] is a well-known trigonometric value:
[tex]\[
\tan\left(\frac{\pi}{3}\right) = \sqrt{3}
\][/tex]
So, when [tex]\( x = \frac{\pi}{3} \)[/tex], the value of the function [tex]\( y = \tan(x) \)[/tex] is [tex]\( \sqrt{3} \)[/tex]. This gives us the point [tex]\( \left(\frac{\pi}{3}, \sqrt{3}\right) \)[/tex].
Next, let's compare it with the given options:
- [tex]\(\left(\frac{\sqrt{3}}{3}, \frac{\pi}{3}\right)\)[/tex]: This point has the x-coordinate [tex]\(\frac{\sqrt{3}}{3}\)[/tex] and y-coordinate [tex]\(\frac{\pi}{3}\)[/tex], which does not match our point [tex]\(\left( \frac{\pi}{3}, \sqrt{3} \right) \)[/tex].
- [tex]\(\left(\frac{\pi}{3}, \frac{\sqrt{3}}{3}\right)\)[/tex]: This point has the y-coordinate [tex]\(\frac{\sqrt{3}}{3}\)[/tex], which is incorrect because our y-coordinate is [tex]\(\sqrt{3} \)[/tex], not [tex]\(\frac{\sqrt{3}}{3}\)[/tex].
- [tex]\(\left(\frac{\pi}{3}, \sqrt{3}\right)\)[/tex]: This point has the coordinates [tex]\(\left( \frac{\pi}{3}, \sqrt{3} \right)\)[/tex], which exactly matches our calculated point.
- [tex]\(\left(\sqrt{3}, \frac{\pi}{3}\right)\)[/tex]: In this case, the x-coordinate is [tex]\(\sqrt{3}\)[/tex] and the y-coordinate is [tex]\(\frac{\pi}{3}\)[/tex], which does not match our point [tex]\(\left( \frac{\pi}{3}, \sqrt{3} \right) \)[/tex].
Therefore, the correct point that the graph of the parent function [tex]\( y = \tan(x) \)[/tex] passes through is:
[tex]\[
\boxed{\left(\frac{\pi}{3}, \sqrt{3}\right)}
\][/tex]
