Problem 106 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Draw the graph of the function $$ \begin{aligned} F(x) &=x-[x], & & 2 n \leq x<2 n+1 \\ &=\frac{1}{2} \quad, & & 2 n+1 \leq x<2 n+2 \end{aligned} $$ where \(n\) is an integer. Is the function periodic? If periodic, what is its period? What are the points of discontinuity of \(F(x) ?\\{\)

Step-by-Step Solution

Answer

Yes, the function is periodic with a period of 2 units. The points of discontinuity are at x = 2n+1 where n is an integer.

Step 1: Setting up the Graph

We start by setting up the graph. Given the nature of our function, it's important to note that the function is defined differently at intervals of (2n to 2n+1) and (2n+1 to 2n+2). Here, n refers to an integer. We'll start by plotting the different sections of the function piece by piece, starting with the range (2n to 2n+1).

Step 2: Plotting the Function for the Range (2n to 2n+1)

For this range, the function F(x) = x - [x]. The floor function [x] will always floor down x to the nearest integer. Since we've defined x as existing between 2n and 2n+1, where n is an integer, [x] will round down to 2n. Thus the function in this range is a straight line with a positive slope.

Step 3: Plotting the Function for the Range (2n+1 to 2n+2)

Here, the function is defined as being constantly equal to 1/2. Hence, in this range, the graph of the function is a straight line parallel to the x axis.

Step 4: Analyzing Periodicity and Points of Discontinuity

Looking at the pattern being repeated across the graph of the function, we can see that the function is indeed periodic. Here, the period is every two units along the x axis. The points of discontinuity occur when x is an odd integer (2n+1), where n is an integer. At these points, the function doesn’t continue smoothly but instead makes a jump from one level to another.

Key Concepts

Periodicity of Functions

Periodicity in functions is a fundamental concept in mathematics, particularly relevant when graphing piecewise functions like the one presented in the exercise. A function is said to be periodic if it repeats its values at regular intervals, known as the 'period'. These intervals can be found by adding or subtracting the period to the function's input without changing its output. For example, the sine function, \(\sin(x)\), has a period of \(2\pi\) because \(\sin(x+2\pi) = \sin(x)\) for all values of x.

When looking at the given function, \(F(x)\), you will notice the graph repeats its pattern with each interval that spans from one even integer to the next. Specifically, in this example, the function exhibits a repeating pattern every two units along the x axis, indicating that the function has a period of 2. Understanding periodicity is crucial for correctly drawing the function over an extended interval and predicting its behavior. The significance of recognizing a function's period cannot be overstated, as it allows for easier function analysis and graphing, especially when dealing with complex, real-world phenomena like waves and vibrations.

Points of Discontinuity

Points of discontinuity in a function are locations where the function does not have a well-defined value, or the graph of the function makes 'jumps' from one value to another abruptly. In essence, these are spots where the function 'breaks.' Discontinuities are important to identify, as they can affect the behavior and properties of the function, such as limits and integrability.

In the case of our piecewise function \(F(x)\), discontinuities occur at odd integer values of x, where the function transitions from \(F(x) = x - \lfloor x \rfloor\) to \(F(x) = \frac{1}{2}\). At these points, namely \(x = 2n+1\) where n is an integer, there's a sudden jump in the function's value. Discontinuities are critical to graph correctly, as they are places where you must 'lift your pencil off the paper' when drawing the function. Recognizing and plotting these discontinuities provide a clearer and more accurate representation of the function's behavior.

Floor Function

The floor function is a mathematical function denoted as \(\lfloor x \rfloor\) and is also known as the 'greatest integer function.' It maps a real number to the largest integer less than or equal to it. For instance, \(\lfloor 2.9 \rfloor = 2\) and \(\lfloor -3.1 \rfloor = -4\). The graph of the floor function takes the shape of 'steps' which are why it's often used in piecewise functions to create these step patterns.

In our given function, the floor function appears in the segment \(F(x) = x - \lfloor x \rfloor\), for ranges \(2n \leq x < 2n+1\). Here, as x varies within the specified range, \(\lfloor x \rfloor\) remains constant because x will never reach the next integer. Consequently, the graph of this section will be a line that ascends diagonally until the next step at \(x = 2n+1\), where it drops down to create the 'step.' It's critical to understand how the floor function behaves to graph such functions properly, as it determines where the 'steps' occur and clearly outlines the intervals of activity within the piecewise function.