Problem 159 - A Complete Resource Book in Mathematics for JEE Main 2017

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 function is one-one and onto when \(B = \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\), so the answer is (D).

Step 1: Recognize the Function

The function given is \(f(x) = \tan^{-1}\left( \frac{2x}{1-x^2} \right)\). This is a form of the inverse tangent function, which typically has a range of \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\). We need to analyze when this function is both one-to-one and onto.

Step 2: Identify the Range

The expression \(\frac{2x}{1-x^2}\) in \(f(x)\) is derived from the transformation of a hyperbolic form. As \(x\) approaches the limits of its domain \((-1,1)\), this term tends towards \(\infty\) and \(-\infty\). This transformation keeps \(f(x)\) in the range of the arctangent function, \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\).

Step 3: Verify One-to-One Property

To be one-to-one, a function must never map two different inputs to the same output. Since \(\tan^{-1}\left( \frac{2x}{1-x^2} \right)\) is derived from the tangent function, and given its monotonic behavior over the interval, it is one-to-one on \((-1,1)\).

Step 4: Confirm Onto Property

The function \(f(x) = \tan^{-1}\left( \frac{2x}{1-x^2} \right)\), covers all possible values in its range \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\) as \(x\) spans \((-1,1)\). It approaches both endpoints of the range as \(x\) approaches the endpoints of its domain.

Step 5: Determine the Correct Interval for B

Given the exploration above, \(f(x)\) is both one-to-one and onto when \(B = \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\). This matches option \(D\).

Key Concepts

One-to-One Functions

A one-to-one function, sometimes called an injective function, has a unique quality: it assigns each input a different output.
This means no two different inputs will ever give you the same output. Think of it like assigning a different locker to each student in a school, ensuring everyone's books are stored separately.

For a function like \( f(x) = \tan^{-1}\left( \frac{2x}{1-x^2} \right) \), understanding this involves examining its behavior on the interval \((-1, 1)\).
- The function is based on the arctangent which is known for its strict monotonic behavior. This means as you increase \(x\), \(f(x)\) also increases, without descending or flattening out after any progression.
- Such monotonic qualities ensure that \(f(x)\) is one-to-one over the interval because it doesn’t turn back or double up on itself.

When solving problems involving one-to-one functions, always consider the graph or the derivative if applicable, as these tools confirm monotonic behavior and thus the function's one-to-one nature.

Onto Functions

Onto functions, or surjective functions, are those that cover every possible value in the target set, meaning the function fills the entire space it's meant to map to.
In the case of \(f(x) = \tan^{-1}\left( \frac{2x}{1-x^2} \right)\), this involves mapping all possible outputs within the given range \((-\frac{\pi}{2}, \frac{\pi}{2})\) as the variable \(x\) crosses its domain from \(-1\) to \(1\).
- Essentially, this means for every y-value in \(B\), there exists at least one x-value in \(A\) such that \(f(x) = y\).
- To determine if a function like \(f(x)\) is onto, check that as \(x\) approaches the endpoints of its domain, \(f(x)\) approaches the extremes of the range (\(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\)).

In practical terms, when we say "f is onto," we mean that f exhausts the entire interval \((-\frac{\pi}{2}, \frac{\pi}{2})\) without leaving gaps, making every element of \(B\) attainable.

Range of a Function

The range of a function encompasses all the possible outputs that the function can produce.
Understanding the range means knowing how wide the set of outputs for a function can extend.

Considering \(f(x) = \tan^{-1}\left( \frac{2x}{1-x^2} \right)\), which is based on the inverse tangent function, its range is typically aligned with the set of real numbers \((-\frac{\pi}{2}, \frac{\pi}{2})\).
- Since the arctangent is symmetrical and expands infinitely within its range, reaching both its upper and lower bounds, the transformation within \(\frac{2x}{1-x^2}\) preserves this dynamic, effectively spanning the arctan's full range as \(x\) transverses the domain \((-1,1)\).
- The domain affects how the transformation behaves, shaping the curve of \(f(x)\) to adapt smoothly from one boundary to the other without divergence.

By knowing just how far and wide a function can go, students grasp the full picture of the function's potential outputs.