The graph shown appears to be of the cosecant function, $f(x) = \csc(x)$.

Here's why:

1. Shape of the Graph: The graph has vertical asymptotes and a repeating wave-like structure. The curves exhibit behavior typical of reciprocal trigonometric functions, specifically resembling the graph of $\csc(x)$ or $\sec(x)$.

2. Position of Asymptotes: The asymptotes occur at multiples of $\pi$, such as $x = -\pi$, $x = 0$, and $x = \pi$. This placement is characteristic of the cosecant function, $\csc(x)$, which has vertical asymptotes at multiples of $\pi$, where the sine function is zero.

3. Symmetry and Periodicity: The graph is symmetric about the origin, which aligns with the odd function property of $\csc(x) = \frac{1}{\sin(x)}$, and it repeats every $2\pi$, matching the period of the cosecant function.

So, the correct function for this graph is: $f(x) = \csc(x)$

Would you like a detailed explanation of the properties of the cosecant function or have any further questions?

Related Questions:

1. What is the period of the cosecant function?
2. How does the graph of $\csc(x)$ compare to $\sec(x)$?
3. What are the domain and range of the $\csc(x)$ function?
4. How can we derive the vertical asymptotes of $\csc(x)$?
5. What are the transformations that can affect the graph of $\csc(x)$?

Tip:

Remember, the cosecant function, being the reciprocal of the sine function, has undefined values wherever sine equals zero, resulting in its characteristic vertical asymptotes.