Problem 72 - A Complete Resource Book in Mathematics for JEE Main

Question

The period of the function \(f(x)= \begin{cases}1, & \text { when } x \text { is a rational } \\ 0, & \text { when } x \text { is irrationa }\end{cases}\) (A) 1 (B) 2 (C) non-periodic (D) None of these

Step-by-Step Solution

Answer

The function is non-periodic (Option C).

Step 1: Understanding the Function

The function \( f(x) \) is a piecewise function defined such that it outputs 1 when \( x \) is rational and 0 when \( x \) is irrational.

Step 2: Defining Periodicity

A function \( f(x) \) is said to be periodic with period \( T \) if \( f(x + T) = f(x) \) for all values of \( x \).

Step 3: Checking Rational Input Case

For any rational number \( x \), \( f(x) = 1 \). If we add any period \( T \) to \( x \), \( x + T \) needs to be rational as well for \( f(x + T) = f(x) \).

Step 4: Checking Irrational Input Case

For any irrational number \( x \), \( f(x) = 0 \). If we add any period \( T \) to \( x \), \( x + T \) needs to be irrational as well for \( f(x + T) = f(x) \).

Step 5: Determining the Periodicity

The key problem is finding a single \( T \) such that \( f(x + T) = f(x) \) for both rational and irrational numbers, but due to the nature of rational and irrational numbers (addition of a real number to a rational number can be irrational and vice versa), no such \( T \) exist.

Step 6: Concluding the Periodicity

Since no \( T > 0 \) exists such that \( f(x + T) = f(x) \) for all \( x \), the function is non-periodic.

Key Concepts

Piecewise Functions

Piecewise functions are special types of functions where different formulas are used to evaluate a function for different intervals of the input variable, often depending on conditions. In other words, a piecewise function is a function composed of multiple sub-functions, each responsible for a specific piece of the domain.
These functions can be written using a brace to group multiple expressions, where each expression has an associated condition. For example, in the function given in the original exercise, the output is determined based on whether the input is rational or irrational.

• If the input is a rational number, the function returns 1.
• If the input is an irrational number, it returns 0.

This approach allows the function to handle different cases within one overall function structure, making it versatile for complex equations that have segments behaving differently.

Rational Numbers

Rational numbers are numbers that can be expressed as a fraction, where both the numerator and the denominator are integers, and the denominator is not zero.
These numbers are dense in the real number line, meaning between any two real numbers, there are infinitely many rational numbers. The concept of rational numbers is grounded in the idea that anything expressible as ratios of whole numbers is a rational number.

• Examples of rational numbers include:
  • 0.75 (which is \(\frac{3}{4}\))
  • -2 (considered as \(\frac{-2}{1}\))
  • 5/7 (a simple fraction)

Rational numbers play a crucial role in mathematics as they provide a bridge between whole numbers and more complex numbers like irrationals and real numbers.

Irrational Numbers

Irrational numbers are numbers that cannot be written as a simple fraction, meaning they cannot be expressed as the ratio of two integers. Their decimal representation is non-repeating and non-terminating.
Irrational numbers fill in the gaps between rational numbers on the number line. Well-known examples include numbers like \( \pi \) (pi) and \( \sqrt{2} \) (the square root of 2), both of which have infinite, non-repeating decimal expansions.

• Characteristics of irrational numbers include:
  • They go on forever without repeating.
  • They cannot be precisely represented as fractions.

Understanding that irrational numbers cannot be rearranged into a fractional form highlights their unique place in mathematics. They coexist with rational numbers to form the set of real numbers, a complete continuum of possible values on the number line.