The function $f(x) = \csc(x)$ represents the cosecant function, which is the reciprocal of the sine function. In mathematical terms:

$\csc(x) = \frac{1}{\sin(x)}$

Domain and Range:

• Domain: The function is undefined where $\sin(x) = 0$, which occurs at integer multiples of $\pi$, i.e., $x = n\pi$, where $n \in \mathbb{Z}$. Therefore, the domain is:

$x \in \mathbb{R} \setminus \{ n\pi \mid n \in \mathbb{Z} \}$

• Range: Since $\csc(x)$ is the reciprocal of $\sin(x)$, and $\sin(x)$ varies between -1 and 1 (but not equal to 0), the range is:

$(-\infty, -1] \cup [1, \infty)$

Key Properties:

• Periodicity: The cosecant function has a period of $2\pi$, which means $\csc(x + 2\pi) = \csc(x)$.
• Asymptotes: Vertical asymptotes occur where $\sin(x) = 0$, at $x = n\pi$.

Would you like more details on the graph of this function or other properties?

Related Questions:

1. How does the graph of $f(x) = \csc(x)$ compare to that of $f(x) = \sin(x)$?
2. Can you explain the behavior of the function near its vertical asymptotes?
3. What is the derivative of $f(x) = \csc(x)$?
4. What is the integral of $f(x) = \csc(x)$?
5. How would $\csc(x)$ behave if it were composed with another function?

Tip:

Always be cautious with reciprocal functions like cosecant, as they introduce vertical asymptotes where the original sine function is zero.