Problem 104 - A Complete Resource Book in Mathematics for JEE Main

Question

The range of the function \(y=\frac{x-[x]}{1-[x]+x}\) (A) \(\left(0, \frac{1}{2}\right)\) (B) \(\left[0, \frac{1}{2}\right]\) (C) \(\left[0, \frac{1}{2}\right)\) (D) \(\left(0, \frac{1}{2}\right]\)

Step-by-Step Solution

Answer

The range of the function is option (C) \([0, \frac{1}{2})\).

Step 1: Understand the Function

The function given is \( y = \frac{x - [x]}{1 - [x] + x} \), where \([x]\) denotes the floor function of \(x\) (also known as the greatest integer function). This function returns the largest integer less than or equal to \(x\). We need to determine the range of \(y\).

Step 2: Simplify the Expression

Rewrite the function as \( y = \frac{x - n}{1 - n + x} \), where \( n = [x] \). Note that \( x - n \) represents the fractional part of \( x \), which satisfies \( 0 \leq x - n < 1 \). Consequently, the denominator is \( 1 - n + x = 1 + (x - n) \).

Step 3: Analyze the Fractional Part

Since the fractional part \( x - n \) ranges from 0 to just below 1, both the numerator \( x - n \) and the denominator \( 1 + (x - n) \) vary accordingly. The denominator will range from 1 to just under 2. The goal is to find the range of the expression \( \frac{x-n}{1+(x-n)} \).

Step 4: Evaluate the Bounds

Evaluate the expression when \(x - n = 0\): \(y = \frac{0}{1} = 0\). When \(x - n\) approaches 1, \(y = \frac{1}{2}\). Thus, the expression approaches but does not reach \(\frac{1}{2}\).

Step 5: Determine the Range

From the analysis, the smallest value of \(y\) is 0 and the largest is approaching but not including \(\frac{1}{2}\). This indicates that \(y\) can take any value in the interval \([0, \frac{1}{2})\).

Step 6: Identify the Correct Option

The correct range for the function \( y = \frac{x - [x]}{1 - [x] + x} \) is the half-open interval \([0, \frac{1}{2})\). Match this with the given options to conclude that option (C) is correct.

Key Concepts

Fractional part

The fractional part of a number is an interesting concept. It represents the part of the number that is leftover after removing the integer part. For any real number \( x \), its fractional part is denoted as \( x - [x] \), where \([x]\) is the greatest integer less than or equal to \( x \). This means:

• If \( x = 3.75 \), then the fractional part is \( 3.75 - 3 = 0.75 \).
• For \( x = 6.2 \), the fractional part is \( 6.2 - 6 = 0.2 \).
• When \( x \) is an integer, like \( x = 4 \), the fractional part is \( 0 \).

The fractional part is always a number between 0 (inclusive) and 1 (exclusive). This is because the integer part completely captures the integer portion of the number. Understanding this can help in analyzing expressions where the fractional part plays a role, like in rational expressions.

Greatest integer function

The greatest integer function, also known as the floor function, acts by rounding a real number down to the nearest integer. It is represented by \([x]\). In essence, this function gives us the highest integer that is less than or equal to \( x \). Let's see how it operates:

• For \( x = 5.9 \), \( [x] = 5 \).
• If \( x = -2.7 \), \( [x] = -3 \) (since \(-3\) is less than but closest to \(-2.7\)).
• And for \( x = 4 \), \( [x] = 4 \) because it is already an integer.

This concept is crucial when working with rational expressions involving the floor function because it affects the form of both the numerator and denominator in a fraction. This can manifest as a constraint on the range of a function, as seen in our given expression where it factors into identifying the possible values \( y \) can take.

Rational expressions

Rational expressions are fractions where the numerator and the denominator are polynomials or other algebraic expressions. In our exercise, the function \( y = \frac{x - [x]}{1 - [x] + x} \) is an example involving the fractional and integer parts.When dealing with rational expressions, one key step is to simplify the expression:

• Simplify the numerator and denominator separately if possible.
• Identify any common factors.

In our function, simplifying the denominator, \( 1 - [x] + x \), gives \( 1 + (x - [x]) \), which uses the fact that \( x - [x] \) is the fractional part. Understanding this is crucial to determining the expression's range. You explore limits where the fractional part goes to 0 or approaches 1, impacting how the entire rational expression behaves. This process shows how specific ranges, like \([0, \frac{1}{2})\), are derived.