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 graphed curve exhibits characteristics consistent with the inverse tangent function ($\tan^{-1} x$ or $\arctan x$). The curve originates from $y = -\frac{\pi}{2}$ for negative $x$ values, displays monotonic increase, and asymptotically approaches $y = \frac{\pi}{2}$ as $x$ approaches infinity. This behavior strongly suggests the inverse tangent function, defined by the following properties:

• The range of the inverse tangent function is established as $-\frac{\pi}{2} \leq y \leq \frac{\pi}{2}$.
• The domain of $y = \tan^{-1} x$ encompasses all real numbers, aligning with the visual representation of the graph.
• The curve is characterized by continuity and smoothness, a common attribute of inverse trigonometric functions.

Consequently, the graph accurately represents $y = \tan^{-1} x$.