The graph shown represents the function $y = \csc(x)$, which is the reciprocal of the sine function, i.e., $\csc(x) = \frac{1}{\sin(x)}$. The graph has vertical asymptotes wherever the sine function is zero, which occurs at multiples of $\pi$, such as $x = -2\pi, -\pi, 0, \pi, 2\pi$, etc.

Now, you are asked to determine certain properties about the inverse of $y = \csc(x)$, that is, $y = \arccsc(x)$. Let's evaluate each of the options:

1. Zeros: None

The cosecant function $\csc(x)$ has no zeros because the sine function is never zero for the values where cosecant is defined (it is undefined at the zeros of sine). Since the inverse function $y = \arccsc(x)$ would also be undefined at zero, there are no zeros. This statement is true.

2. Domain: $(-\infty, -1] \cup [1, \infty)$

The domain of the inverse cosecant function is where the original function $\csc(x)$ takes its values. Since $\csc(x)$ is undefined between -1 and 1 (because sine never reaches those values), the domain of the inverse is $(-\infty, -1] \cup [1, \infty)$. This statement is true.

3. Range: $[0, \pi]$

The range of $\arccsc(x)$ corresponds to the possible angles that yield values within the domain of the cosecant function. For $y = \arccsc(x)$, we are typically concerned with angles in the range $[-\frac{\pi}{2}, \frac{\pi}{2}]$ for the principal value. This statement is false, as the range is usually $[-\frac{\pi}{2}, \frac{\pi}{2}]$, excluding zero.

4. Minimum: $-\frac{\pi}{2}$

The minimum value for the inverse cosecant function occurs when $x = -\infty$, which approaches $-\frac{\pi}{2}$. This statement is true.

5. Maximum: $\frac{\pi}{2}$

The maximum value for the inverse cosecant function occurs when $x = \infty$, which corresponds to $\frac{\pi}{2}$. This statement is true.

6. Increasing on $\left( \frac{\pi}{2}, \pi \right)$ and $\left( \pi, \frac{3\pi}{2} \right)$

The inverse cosecant function is decreasing overall because cosecant decreases as you move along the positive or negative directions from its vertical asymptotes. This statement is false, as the inverse function should not be increasing on these intervals.

Summary of Correct Statements:

• Zeros: None (True)
• Domain: $(-\infty, -1] \cup [1, \infty)$ (True)
• Minimum: $-\frac{\pi}{2}$ (True)
• Maximum: $\frac{\pi}{2}$ (True)

Would you like further details on any part of the explanation? Here are five related questions to help you explore this topic:

1. What is the difference between the graph of $y = \csc(x)$ and $y = \sin(x)$?
2. How does the range of $\arccsc(x)$ compare to other inverse trigonometric functions like $\arcsin(x)$?
3. Why does $\csc(x)$ have vertical asymptotes?
4. Can we define $\csc(x)$ for values between $-1$ and $1$?
5. What are the properties of the graph of $y = \arccsc(x)$?

Tip: When working with inverse functions, always pay attention to both the domain and range of the original function, as they swap roles in the inverse.