To determine which point lies on the graph of the parent function [tex]\( y = \tan(x) \)[/tex], we need to evaluate [tex]\(\tan(x)\)[/tex] at the x-coordinates given in each point and see if it equals the corresponding y-coordinate.
1. First point: [tex]\(\left(\frac{\sqrt{3}}{3}, \frac{\pi}{3}\right)\)[/tex]
Let [tex]\( x = \frac{\sqrt{3}}{3} \)[/tex].
Calculate [tex]\(\tan \left( \frac{\sqrt{3}}{3} \right) \)[/tex].
The result is approximately [tex]\( 0.6513878866881448 \)[/tex].
Since [tex]\( 0.6513878866881448 \neq \frac{\pi}{3} \approx 1.0471975511965976 \)[/tex], this point does not lie on the graph of [tex]\( y = \tan(x) \)[/tex].
2. Second point: [tex]\(\left(\frac{\pi}{3}, \frac{\sqrt{3}}{3}\right)\)[/tex]
Let [tex]\( x = \frac{\pi}{3} \)[/tex].
Calculate [tex]\(\tan \left( \frac{\pi}{3} \right) \)[/tex].
The result is approximately [tex]\( 1.7320508075688767 \)[/tex].
Since [tex]\( 1.7320508075688767 \neq \frac{\sqrt{3}}{3} \approx 0.5773502691896257 \)[/tex], this point does not lie on the graph of [tex]\( y = \tan(x) \)[/tex].
3. Third point: [tex]\(\left(\frac{\pi}{3}, \sqrt{3}\right)\)[/tex]
Let [tex]\( x = \frac{\pi}{3} \)[/tex].
Calculate [tex]\(\tan \left( \frac{\pi}{3} \right) \)[/tex].
The result is approximately [tex]\( 1.7320508075688767 \)[/tex].
Since [tex]\( 1.7320508075688767 = \sqrt{3} \approx 1.7320508075688772 \)[/tex], this point lies on the graph of [tex]\( y = \tan(x) \)[/tex].
4. Fourth point: [tex]\(\left(\sqrt{3}, \frac{\pi}{3}\right)\)[/tex]
Let [tex]\( x = \(\sqrt{3} \)[/tex].
Calculate [tex]\(\tan \left( \sqrt{3} \right) \)[/tex].
The result is approximately [tex]\( -6.147533160296622 \)[/tex].
Since [tex]\( -6.147533160296622 \neq \frac{\pi}{3} \approx 1.0471975511965976 \)[/tex], this point does not lie on the graph of [tex]\( y = \tan(x) \)[/tex].
Therefore, the point that lies on the graph of [tex]\( y = \tan(x) \)[/tex] is [tex]\( \left( \frac{\pi}{3}, \sqrt{3} \right) \)[/tex].
