examine the graph of $f$ given below.\ncomplete the equation of the function:\n$f(x) = ?$ select an answer

Explanation:

Step1: Identify the function type

The graph shows periodic vertical asymptotes and x-intercepts, characteristic of a cotangent function.

Step2: Determine the period

The x-intercepts occur at $0, 3.14, 6.28, \dots$, which correspond to $0, \pi, 2\pi, \dots$. The distance between intercepts is $\pi$. $$P = \pi$$

Step3: Find the coefficient $b$

The period of $\cot(bx)$ is $\frac{\pi}{|b|}$. Since $P = \pi$, we solve for $b$. $$\frac{\pi}{b} = \pi \implies b = 1$$

Step4: Check for vertical asymptotes

The asymptotes are at $x = \pm \frac{\pi}{2}, \pm \frac{3\pi}{2}, \dots$. This matches the standard $\cot(x)$ shifted horizontally.

Step5: Identify the horizontal shift

A standard $\cot(x)$ has an asymptote at $x=0$. This graph has an intercept at $x=0$. Thus, it is shifted by $\frac{\pi}{2}$. $$f(x) = \cot\left(x - \frac{\pi}{2}\right)$$

Step6: Simplify using trigonometric identities

Using the identity $\cot\left(x - \frac{\pi}{2}\right) = -\tan(x)$. $$f(x) = -\tan(x)$$

Answer:

f(x) = -\tan(x)