To determine which points are on the graph of \( y = \tan x \), we need to evaluate the tangent function at the given x-coordinates:
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At \( x = \frac{4\pi}{3} \): \[ \tan\left(\frac{4\pi}{3}\right) = \tan\left(\pi + \frac{\pi}{3}\right) = \tan\left(\frac{\pi}{3}\right) = \sqrt{3} \] The point \( \left(\frac{4\pi}{3}, \sqrt{3}\right) \) is on the graph.
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At \( x = \frac{\pi}{4} \): \[ \tan\left(\frac{\pi}{4}\right) = 1 \] The point \( \left(\frac{\pi}{4}, -1\right) \) is not on the graph because the corresponding y-value should be \( 1 \).
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At \( x = \frac{\pi}{6} \): \[ \tan\left(\frac{\pi}{6}\right) = \frac{1}{\sqrt{3}} \] The point \( \left(\frac{\pi}{6}, -\frac{\sqrt{3}}{3}\right) \) is not on the graph because the corresponding y-value should be \( \frac{1}{\sqrt{3}} \).
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At \( x = \frac{\pi}{3} \): \[ \tan\left(\frac{\pi}{3}\right) = \sqrt{3} \] The point \( \left(\frac{\pi}{3}, \sqrt{3}\right) \) is on the graph.
In summary, the points on the graph of \( y = \tan x \) are:
- \( \left(\frac{4\pi}{3}, \sqrt{3}\right) \)
- \( \left(\frac{\pi}{3}, \sqrt{3}\right) \)
Therefore, the correct responses are:
- \( \left(\frac{4\pi}{3}, \sqrt{3}\right) \)
- \( \left(\frac{\pi}{3}, \sqrt{3}\right) \)
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