Problem 14 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Let \(f(x)=[3+2 \cos x], x \in\left(\frac{-\pi}{2}, \frac{\pi}{2}\right)\), where \([.]\) denotes the greatest integer function. The number of points of discontinuity of \(f(x)\) is (A) 3 (B) 2 (C) 5 (D) None of these

Step-by-Step Solution

Answer

There are 2 points of discontinuity for \( f(x) \).

Step 1: Understanding the Function

The function given is \( f(x) = [3 + 2 \cos x] \), where the bracket \([.]\) indicates the greatest integer function. This function outputs the greatest integer less than or equal to the value inside the bracket.

Step 2: Domain and Range of Cosine Function

Since \( x \in \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) \), within this interval \( \cos x \) ranges from \( 0 \) to \( 1 \). Therefore, \( 2 \cos x \) ranges from \( 0 \) to \( 2 \).

Step 3: Evaluating the Range of \(3 + 2\cos x\)

The expression \( 3 + 2 \cos x \) thus ranges from \( 3 \) to \( 5 \). Consequently, \( f(x) = [3 + 2 \cos x] \) can evaluate to integers 3, 4, or 5.

Step 4: Identifying Points of Discontinuity

The greatest integer function \([a]\) is discontinuous at integer values of \(a\). Thus, we check where \( 3 + 2 \cos x \) takes integer values (since \( x \in \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) \)). These occur when \( 3 + 2 \cos x = 4 \) or \( 4 \) and at boundaries of these values. This leads to discontinuities at these exact points of transition.

Step 5: Counting the Points of Discontinuity

For \( f(x) \) to be discontinuous, the expression must equal an integer, specifically around transition from below an integer to the integer itself. In our case, discontinuity is at \( 3 + 2 \cos x = 4 \) or \( = 5 \). Evaluate \( \cos x = \frac{1}{2}, \frac{0}{2} \) these yield to \( f(x) = 3,4,5 \), pointing out discontinuous transitions occur at boundaries between these integer values.

Step 6: Conclusion on Number of Discontinuities

As \( 2 \cos x \) transitions at points leading \( 3 + 2 \cos x \) at threshold boundaries, continuity holds except exactly at those jumps, we confirm discontinuity at integer shifts giving a total of 2 main transition points for discontinuity.

Key Concepts

Discontinuity in Functions

In mathematics, a discontinuity is where a function is not continuous, meaning it jumps abruptly at certain points. The greatest integer function, denoted as \([x]\), is inherently discontinuous at any integer value of \('x'\). Essentially, this function takes a real number and outputs the greatest integer less than or equal to it. Because of this nature, whenever the output of the function reaches a whole number, the function exhibits a jump discontinuity.
The present case studies a function \(f(x) = [3 + 2 \cos x]\). Here, the expression inside the brackets \(3 + 2 \cos x\) can smoothly range over continuous real outputs. Still, the function itself will leap at specific points as it reaches an integer.
Discontinuities occur when \(3 + 2 \cos x\) becomes an integer, such as 4 or 5. These serve as natural breaking points for the function, distinct intervals where the function's values suddenly shift without covering every real value in-between.

Cosine Function Behavior

The cosine function, \(\cos x\), is known for its smooth oscillations between -1 and 1. In our given range \(x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\), this function behaves particularly neatly since it only spans from 0 to 1. To analyze the composite function \(f(x) = [3 + 2\cos x]\), understanding the behavior of \(\cos x\) is crucial.
The expression \(2 \cos x\) merely stretches the range of \(\cos x\) from the interval \(0, 1\) to \(0, 2\). Consequently, adding 3 shifts this interval to 3 to 5. Throughout these transformations, \(\cos x\) demonstrates its periodic nature as it smoothly fills the available range in its section of the unit circle. Influencing the main function's transition, this steady motion helps identify the exact points where discontinuities occur based on integer jumps.

Domain and Range Analysis

The domain of the function \(f(x) = [3 + 2 \cos x]\) is contained within the interval \(x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\). This restriction directly impacts the behavior of the cosine component within it. Given this domain, \(\cos x\) is restricted to values between 0 and 1.
Next, consider the range transformation. By calculating \(3 + 2 \cos x\), the output extends to a range of potential values from 3 to 5. Thus, the corresponding range of the greatest integer function \(f(x)\) can only be the integers 3, 4, or 5.

• When \(3 + 2 \cos x\) is just below 4, \(f(x) = 3\).
• When exactly 4, \(f(x)\) jumps to 4, and similarly just below 5, it remains 4.
• Finally, it hits 5 when \(3 + 2 \cos x\) equals precisely 5 without exceeding.

Understanding the domain and range helps pinpoint exactly which integer values can result and where discontinuities lie within the function's output range.