Problem 134 - A Complete Resource Book in Mathematics for JEE Main

Question

The period of the function \(f(x)=3 x+3-[3 x+3]+\) \(\sin \frac{\pi x}{2}\), where \([x]\) denotes the greatest integer \(\leq x\), is (A) 4 (B) 1 (C) 2 (D) None of these

Step-by-Step Solution

Answer

The period of the function is 4.

Step 1: Identify parts of the function

The function is given as \( f(x) = 3x + 3 - [3x + 3] + \sin\left(\frac{\pi x}{2}\right) \). It consists of three parts: \(3x + 3\), the greatest integer function \([3x+3]\), and the trigonometric function \(\sin\left(\frac{\pi x}{2}\right)\).

Step 2: Analyze the linear and floor function

The term \(3x + 3 - [3x + 3]\) results in the fractional part of \(3x + 3\), denoted as \(\{3x + 3\}\). This function is periodic with period 1 because the fractional part of a function \(\{x\}\) repeats every 1 unit.

Step 3: Analyze the sine function

The function \(\sin\left(\frac{\pi x}{2}\right)\) has a period \(\frac{2\pi}{\frac{\pi}{2}} = 4\). This means the sine component repeats every 4 units.

Step 4: Find the overall period of the function

To find the overall period of \(f(x)\), identify the least common multiple of the periods of each part: \(\{x\}\) has period 1 and \(\sin\left(\frac{\pi x}{2}\right)\) has period 4. The least common multiple of 1 and 4 is 4. Thus, the function \(f(x)\) has a period of 4.

Key Concepts

Understanding the Greatest Integer Function

The greatest integer function, often symbolized as \[ [x] \], represents the largest integer that is less than or equal to a given number \((x)\). For example, applying \[ [x] \] to 3.6 results in 3, because 3 is the largest integer less than or equal to 3.6. This function is also known as the floor function.

• Discrete Jumps: The function is characterized by a series of jumps as the value of \((x)\) crosses an integer boundary increases, resulting in a step-wise graph.
• Periodicity in Context: When used within an expression such as \[ [3x + 3] \], the function modifies the input expression and can affect the periodicity of the entire function.

Understanding how it modifies the expression is key to analyzing periodic functions such as those involving both floor and trigonometric components.

The Role of Trigonometric Functions

Trigonometric functions, such as sine and cosine, are fundamental in mathematics due to their repetitive wave-like patterns. Specifically, \(\sin\left(\frac{\pi x}{2}\right)\) plays an important role in determining the periodic behavior of functions.

• Wave Characteristics: The sine function naturally oscillates between -1 and 1, creating a smooth, continuous wave.
• Predictable Periods: Typically, the standard \(\sin(x)\) function exhibits a period of \(2\pi\), but modifications, such as multiplying the input by \((\pi/2)\), adjust the frequency of oscillation. This results in a new period of 4 in our specific function example.

Understanding the sine function helps comprehend its impact on composite periodic functions, as it introduces regular intervals of repetition.

Exploring the Fractional Part Function

The fractional part function, denoted by \[ \{x\} = x - [x] \], captures the non-integer component of \((x)\). For instance, if \(x = 3.6\), then \(\{3.6\} = 3.6 - 3 = 0.6\).

• Periodic Nature: This function repeats every unit interval, as any real number between the integers will map into itself and return to its original form after passing another integer.
• Combining Functions: In expressions combining both linear and fractional parts, such as \[ 3x + 3 - [3x + 3] \], the result is the fractional part function, \[ \{3x + 3\} \].
• Contribution to Periodicity: When incorporated into a larger function, its intrinsic period of 1 interacts with other components (like trigonometric parts) to influence the function's overall periodicity.

Understanding how \[ \{x\} \] functions play into larger expressions aids in deciphering complex periodic behaviors.