1. The problem is to create and understand the function $f(x) = \tan x$. 2. The function $\tan x$ is defined as the ratio of sine to cosine: $$\tan x = \frac{\sin x}{\cos x}$$. 3. Important rules: - The function is undefined where $\cos x = 0$, which happens at $x = \frac{\pi}{2} + k\pi$, for any integer $k$. - The function has vertical asymptotes at these points. - The function is periodic with period $\pi$. 4. To analyze the function, consider values in one period, for example $x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$. 5. The function passes through the origin: $f(0) = \tan 0 = 0$. 6. As $x$ approaches $\frac{\pi}{2}^-$, $\tan x \to +\infty$ and as $x$ approaches $\frac{\pi}{2}^+$, $\tan x \to -\infty$. 7. The function is odd: $\tan(-x) = -\tan x$. Final answer: The function is $f(x) = \tan x$ with vertical asymptotes at $x = \frac{\pi}{2} + k\pi$, period $\pi$, and zeros at $x = k\pi$ for integers $k$.