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

Analysis of the graph reveals the following characteristics:
- The graph's domain is \(|x| \geq 1\), with vertical asymptotes at \(x = -1\) and \(x = 1\). This aligns with the properties of \(y = \csc^{-1} x\).
- The function's range is \([0, \frac{\pi}{2}) \cup (\frac{\pi}{2}, \pi]\) and its negative counterpart \([-\pi, -\frac{\pi}{2}) \cup (-\frac{\pi}{2}, 0]\), which are the principal values for the inverse cosecant function.
- The graph is aperiodic, thus excluding periodic trigonometric functions such as \(\csc x\) or \(\sec x\).
Based on these observations, the graph represents the inverse cosecant function, \(y = \csc^{-1} x\).