What is the Graph of y = [x]?

S3-SA5-0095

Grade Level:

Class 9

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

The graph of y = [x] shows how the greatest integer less than or equal to x changes as x changes. It looks like a series of steps, which is why it's also called a 'step function' or 'floor function'. For any input x, the output y is always a whole number.

Simple Example

Quick Example

Imagine you're buying chai, and the shopkeeper always rounds down the price to the nearest whole rupee. If a chai costs Rs 10.75, you pay Rs 10. If it costs Rs 10.00, you pay Rs 10. If it costs Rs 10.99, you still pay Rs 10. This rounding down is exactly what the [x] function does.

Worked Example

Step-by-Step

Let's plot some points for y = [x] to understand its graph.

Step 1: Consider values of x between 0 and 1 (but not including 1).
If x = 0.1, y = [0.1] = 0.
If x = 0.5, y = [0.5] = 0.
If x = 0.99, y = [0.99] = 0.
So, for 0 <= x < 1, y is always 0. We draw a horizontal line segment from (0,0) to (1,0) with a closed circle at (0,0) and an open circle at (1,0).

---

Step 2: Consider values of x between 1 and 2 (but not including 2).
If x = 1.0, y = [1.0] = 1.
If x = 1.5, y = [1.5] = 1.
If x = 1.99, y = [1.99] = 1.
So, for 1 <= x < 2, y is always 1. We draw a horizontal line segment from (1,1) to (2,1) with a closed circle at (1,1) and an open circle at (2,1).

---

Step 3: Consider values of x between 2 and 3 (but not including 3).
If x = 2.0, y = [2.0] = 2.
If x = 2.5, y = [2.5] = 2.
If x = 2.99, y = [2.99] = 2.
So, for 2 <= x < 3, y is always 2. We draw a horizontal line segment from (2,2) to (3,2) with a closed circle at (2,2) and an open circle at (3,2).

---

Step 4: Consider values of x between -1 and 0 (but not including 0).
If x = -0.9, y = [-0.9] = -1.
If x = -0.5, y = [-0.5] = -1.
If x = -0.01, y = [-0.01] = -1.
So, for -1 <= x < 0, y is always -1. We draw a horizontal line segment from (-1,-1) to (0,-1) with a closed circle at (-1,-1) and an open circle at (0,-1).

---

Answer: The graph of y = [x] consists of horizontal line segments, each of length 1 unit, starting at a whole number on the x-axis and ending just before the next whole number. Each segment has a closed circle at its left end and an open circle at its right end, forming steps.

Why It Matters

Understanding step functions like y = [x] is crucial in Computer Science for things like memory allocation or timing events, where values are often rounded down. In Data Science, it helps in processing discrete data. Engineers use it in digital signal processing to convert continuous signals into discrete ones.

Common Mistakes

MISTAKE: Thinking that [x] rounds to the nearest integer. For example, [-0.5] = 0. | CORRECTION: [x] always rounds DOWN to the greatest integer less than or equal to x. So, [-0.5] = -1, because -1 is the greatest integer less than or equal to -0.5.

MISTAKE: Drawing a continuous line or a diagonal line for the graph. | CORRECTION: The graph of y = [x] is NOT continuous. It jumps at every integer value of x, creating distinct horizontal 'steps'.

MISTAKE: Confusing the closed and open circles on the graph. For example, drawing a closed circle at (1,1) and (2,1). | CORRECTION: The left end of each step (at an integer x-value) has a closed circle because y = [x] includes that integer value. The right end of each step (just before the next integer x-value) has an open circle because the function jumps to the next integer output at that point.

Practice Questions

Try It Yourself

QUESTION: What is the value of y when x = 3.8 in y = [x]? | ANSWER: 3

QUESTION: For what values of x will the graph of y = [x] show y = -2? | ANSWER: -2 <= x < -1

QUESTION: Sketch the graph of y = [x] for x values from -2 to 2. Label the points where the function jumps. | ANSWER: The graph will show steps: a segment at y=-2 for -2<=x<-1, a segment at y=-1 for -1<=x<0, a segment at y=0 for 0<=x<1, and a segment at y=1 for 1<=x<2. Jumps occur at x = -1, 0, 1, 2.

MCQ

Quick Quiz

Which of the following describes the graph of y = [x]?

A continuous straight line

A series of diagonal lines

A series of horizontal steps with jumps at integer values

A curve that goes up and down

The Correct Answer Is:

The graph of y = [x] is a step function. It remains constant (horizontal) for a range of x-values and then 'jumps' to the next integer value at every integer x. Options A, B, and D describe different types of graphs.

Real World Connection

In the Real World

Imagine a mobile network provider that charges you for data in full 1 GB blocks. Even if you use 0.1 GB, you're charged for 1 GB. If you use 1.5 GB, you're charged for 2 GB. This is similar to a ceiling function, but the 'floor' function [x] is used in many digital systems, like how a digital clock displays time (it shows 2:00 until it's exactly 3:00, not 2:59.99).

Key Vocabulary

Key Terms

Greatest Integer Function: A function that gives the greatest integer less than or equal to the input x. | Step Function: A function whose graph resembles a series of steps. | Floor Function: Another name for the greatest integer function, because it 'rounds down'. | Discontinuous: A graph that has breaks or jumps.

What's Next

What to Learn Next

Great job understanding the graph of y = [x]! Next, explore the graph of y = {x}, which is the fractional part function. It builds on what you've learned about [x] and will show you another interesting type of graph!