Problem 159 - A Complete Resource Book in Mathematics for JEE Main

Question

Let \(f:(-1,1) \rightarrow B\), be a function defined by \(f(x)=\tan ^{-1} \frac{2 x}{1-x^{2}}\), then \(f\) is both one-one and onto when \(B\) is the interval (A) \(\left(0, \frac{\pi}{2}\right)\) (B) \(\left[0, \frac{\pi}{2}\right)\) (C) \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\) (D) \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\)

Step-by-Step Solution

Answer

The correct option is (D) \((-\frac{\pi}{2}, \frac{\pi}{2})\).

Step 1: Identify the Range of Inverse Function

The function given is \( f(x) = \tan^{-1} \left( \frac{2x}{1-x^2} \right) \). Since it's an inverse function, the range of \( \tan^{-1} \theta \) is \( \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \).

Step 2: Determine the Domain of the Argument

The argument of the \( \tan^{-1} \) function is \( \frac{2x}{1-x^2} \). For this function, \(-1 < x < 1\) ensures that the denominator \(1-x^2\) is positive.

Step 3: Analyze the Behavior of \(\frac{2x}{1-x^2}\)

As \(x\) moves from \(-1\) to 1, \(\frac{2x}{1-x^2}\) spans from \(-\infty\) to \(\infty\). Therefore, every real number is achievable as \( \frac{2x}{1-x^2} \).

Step 4: Match the Range of \(f(x)\) with the Interval Options

Based on the above, the range of \(f(x)\) corresponds to the maximum achievable range of \(\tan^{-1}\), which is \( \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \).

Step 5: Select Correct Option for Interval \(B\)

The interval in which \( f(x) \) is both one-one and onto is \( \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \), matching option (D).

Key Concepts

One-One and Onto Functions

Understanding One-One (Injective) and Onto (Surjective) functions is crucial to grasping how functions behave. A function is called **one-one** if it assigns distinct outputs for distinct inputs. In other words, no two different inputs produce the same output. This ensures that each element in the domain maps to a unique element in the codomain.
**Onto** functions, on the other hand, mean that every element in the codomain is the result of the function using some element from the domain. Thus, the function covers the whole codomain, leaving no gaps.
In any context involving trigonometric functions, particularly inverse ones, ensuring a function is both one-one and onto is vital to determine its range and applicability. Importantly, inverse trigonometric functions often naturally fulfill these properties over specific intervals, which is carefully examined when determining function characteristics.

Range of Functions

The range of a function is all the possible output values it can produce. For inverse trigonometric functions like \(\tan^{-1}\), the range is typically defined by the known behavior of the function's inverse across its domain. The function \(f(x) = \tan^{-1} \left( \frac{2x}{1-x^2} \right)\) takes advantage of the full range of \(\tan^{-1}\), which is \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\).
This range includes all angles the inverse tangent function can achieve, aligning with all real numbers due to the property of inverse tangent. Thus, the function's range covers a vast span, matching choice (D) from the exercise. This concept of range is crucial to understanding why certain intervals qualify as both one-one and onto.

Domain of Functions

The domain of a function is the set of all input values for which the function is defined. For the function \(f(x) = \tan^{-1} \left( \frac{2x}{1-x^2} \right)\), the domain is \((-1, 1)\). This specific domain is chosen to ensure that the denominator in the argument, \(1-x^2\), remains positive, preventing division by zero and ensuring a valid input for the tangent function.
Choosing an appropriate domain for trigonometric functions ensures they behave predictably without undefined expressions. Additionally, this interval specifically captures all significant variations of the function, allowing it to span \(-\infty\) to \(\infty\) through \(\frac{2x}{1-x^2}\). Thus, understanding the domain helps to define how the function operates and what range it is capable of achieving.