Question

The graph shown below depicts:

• A. $y = \sec^{-1} x$
• B. $y = \sec x$
• C. $y = \csc^{-1} x$
• D. $y = \csc x$

Hint:

Graphs of inverse trigonometric functions are non-periodic and have restricted domains. The domain of $y = \csc^{-1} x$ is $x \leq -1$ or $x \geq 1$, and its range is $\left[-\frac{\pi}{2}, 0\right) \cup \left(0, \frac{\pi}{2}\right]$.

Answer 1

The Correct Option is C

Let's analyze the features of the graph shown:
- The graph has a range from $[0, \frac{\pi}{2}) \cup (\frac{\pi}{2}, \pi]$ and similar behavior mirrored across the X-axis (i.e., also in $[-\pi, -\frac{\pi}{2}) \cup (-\frac{\pi}{2}, 0]$), which corresponds to the principal values of the inverse cosecant function. 
- The curve is defined for $|x| \geq 1$ and has vertical asymptotes at $x = -1$ and $x = 1$, which is consistent with $y = \csc^{-1} x$. 
- The graph is not periodic, which rules out trigonometric functions like $\csc x$ or $\sec x$, which are periodic. 
Therefore, this graph corresponds to the inverse cosecant function: $y = \csc^{-1} x$.