Problem 76 - A Complete Resource Book in Mathematics for JEE Main

Question

Let \(f(x)\) be a function defined on \([0,1]\) such that $$ f(x)= \begin{cases}x & x \in Q \\ 1-x & x \notin Q\end{cases} $$ Then, for all \(x \in[0,1], f o f(x)\) is (A) a constant (B) \(1+x\) (C) \(x\) (D) None of these

Step-by-Step Solution

Answer

For all \( x \in [0,1] \), \( f \circ f(x) = x \) (Answer: Option C).

Step 1: Understand the Function Definition

The function \( f(x) \) is defined piecewise. When \( x \) is a rational number (\( x \in Q \)), \( f(x) = x \). When \( x \) is an irrational number (\( x otin Q \)), \( f(x) = 1 - x \). We need to find \( f(f(x)) \) for any \( x \) in \([0, 1]\).

Step 2: Evaluate \( f(f(x)) \) for Rational \( x \)

When \( x \in Q \), \( f(x) = x \). Now, we substitute \( f(x) \) back into the function. Since \( x \in Q \), it follows that \( f(x) = x \), so we have \( f(f(x)) = f(x) = x \).

Step 3: Evaluate \( f(f(x)) \) for Irrational \( x \)

When \( x otin Q \), \( f(x) = 1 - x \). Now, 1 - x is a rational number because any irrational subtracted from 1 is rational. Thus, now \( f(x) \) is a rational number, and we use the property \( f(x) = x \) for rational numbers. Therefore, \( f(f(x)) = f(1-x) = 1-x \). Since \( 1-x otin Q \) when \( xotin Q \), we indeed have \( f(f(x)) = x \) by checking that \( f(f(x)) = f(1-x) = x \).

Step 4: Conclusion Whether for Rational or Irrational

No matter if \( x \) is rational or irrational, upon calculating \( f(f(x)) \), we always end up with \( x \). Thus, \( f(f(x)) = x \) for all \( x \) in \([0, 1]\).

Key Concepts

Rational Numbers

Rational numbers are essential in mathematics, known as numbers that can be expressed as the ratio of two integers. If you can represent a number as \( \frac{a}{b} \), where \( a \) and \( b \) are integers and \( b eq 0 \), it is a rational number.
Examples of rational numbers include fractions like \( \frac{1}{2} \), \( \frac{3}{4} \), and whole numbers such as \( 5 \) and \( -2 \).

• Characteristics of Rational Numbers:
  • Can be positive, negative, or zero.
  • Decimal form is either terminating or repeating.
  • Includes integers, fractions, and mixed numbers.
• Why Rational Numbers Behave This Way:
  • The part of rational numbers that makes them intriguing is their ability to seamlessly fit into various mathematical operations and properties, like closure, associativity, and distribution.

In the context of piecewise functions like \( f(x) \) from the exercise, rational numbers allow predictable outputs, essential when evaluating piecewise functions for different input types.

Irrational Numbers

Irrational numbers are those which cannot be expressed as a simple fraction. Unlike rational numbers, their decimal form goes on forever without repeating. This unpredictability is what sets them apart and what can sometimes make them tricky to work with in piecewise functions. Examples include \( \pi \) and \( \sqrt{2} \).
Here are some key points about irrational numbers:

• Decimal Expansion: Non-terminating and non-repeating.
• Cannot be expressed as \( \frac{a}{b} \) where \( a \) and \( b \) are integers.
• They often arise in geometry, such as the diagonal of a square or the circumference of a circle divided by its diameter.
• Not "normal numbers" but they complete the real number line when combined with rational numbers.

In our exercise, when \( x \) is irrational, the piecewise function \( f(x) = 1 - x \) cleverly exploits the inherent nature of irrational numbers, leading to a rational outcome, hence allowing the function to flip its behavior.

Function Composition

Function composition involves taking the output of one function and using it as the input for another. In mathematical terms, if you have two functions \( f \) and \( g \), the composition is represented as \( f(g(x)) \). This concept allows for complex transformations and operations.Let's break down some key elements of function composition:

• Order Matters: The order of functions in composition is crucial. \( f(g(x)) \) is generally not the same as \( g(f(x)) \).
• Multiple Transformations: Composition lets you apply several operations in sequence, building upon each result.
• Domain and Range: The outcome of composing functions depends heavily on the overlap of their domains and ranges, requiring careful consideration of input and output capabilities.

In the exercise, composing the function \( f \) with itself, \( f(f(x)) \), revealed a self-same result \((x)\) regardless of the nature (rational or irrational) of the initial input. This illustrates one powerful aspect of function composition, maintaining a consistent transformation across diverse types of inputs.