Question

The given graph illustrates:

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

Hint:

The graph of $y = \tan^{-1} x$ (inverse tangent) has horizontal asymptotes at $y = \pm \frac{\pi}{2}$ and passes through the origin. Keep these properties in mind while identifying similar graphs.

Answer 1

The Correct Option is A

The graph shows a curve that is characteristic of the inverse tangent function ($\tan^{-1} x$ or $\arctan x$). The curve starts from $y = -\frac{\pi}{2}$ when $x$ is negative, increases monotonically, and approaches $y = \frac{\pi}{2}$ as $x$ tends to infinity. This behavior is a clear indicator of the inverse tangent function, which has the following properties:

• The range of the inverse tangent function is $-\frac{\pi}{2} \leq y \leq \frac{\pi}{2}$.
• The domain of $y = \tan^{-1} x$ is all real numbers, which matches the graph's shape.
• The curve is continuous and smooth, typical of inverse trigonometric functions.

Thus, the graph corresponds to $y = \tan^{-1} x$.