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What is a Closed Interval on a Graph?
Grade Level:
Class 9
AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering
Definition
What is it?
A closed interval on a graph represents a set of all real numbers between two specific points, including both those specific points themselves. It is shown using square brackets, like [a, b], where 'a' is the starting point and 'b' is the ending point.
Simple Example
Quick Example
Imagine your school sets marks for a test. If the marks can be anywhere from 0 to 100, including 0 and 100, then the range of marks is a closed interval. We would write this as [0, 100].
Worked Example
Step-by-Step
Let's plot the closed interval [2, 7] on a number line.
STEP 1: Draw a straight horizontal line. This is your number line.
---STEP 2: Mark a few numbers on the line, like 0, 1, 2, 3, 4, 5, 6, 7, 8.
---STEP 3: Locate the starting point, which is 2. Draw a solid (filled-in) circle at the number 2 on your number line. This solid circle shows that 2 is included in the interval.
---STEP 4: Locate the ending point, which is 7. Draw another solid (filled-in) circle at the number 7 on your number line. This solid circle shows that 7 is also included.
---STEP 5: Draw a thick line or shade the region between the solid circle at 2 and the solid circle at 7. This shaded region represents all the numbers between 2 and 7.
---ANSWER: The shaded line segment, including the solid circles at 2 and 7, represents the closed interval [2, 7].
Why It Matters
Understanding closed intervals is crucial for many advanced subjects. In Data Science, it helps define valid ranges for data, like the age group for a survey. Engineers use it to specify safe operating temperatures for machines, ensuring they work within a defined, inclusive range.
Common Mistakes
MISTAKE: Using round brackets ( ) instead of square brackets [ ] for a closed interval. | CORRECTION: Always use square brackets [ ] to show that the endpoints are included in the interval.
MISTAKE: Drawing open circles at the endpoints on a number line for a closed interval. | CORRECTION: For a closed interval, always draw solid (filled-in) circles at the endpoints to indicate they are included.
MISTAKE: Confusing the order and writing [b, a] instead of [a, b] when 'a' is smaller than 'b'. | CORRECTION: Always write the smaller number first and the larger number second, like [smaller number, larger number].
Practice Questions
Try It Yourself
QUESTION: Represent the closed interval [-3, 5] on a number line. | ANSWER: Draw a number line. Place a solid circle at -3 and another solid circle at 5. Shade the line segment between these two solid circles.
QUESTION: Which of these numbers are included in the closed interval [10, 20]: 9, 10, 15.5, 20, 21? | ANSWER: 10, 15.5, 20
QUESTION: A mobile app's 'Age Group' setting allows users from 13 years to 18 years, including both 13 and 18. Write this age group as a closed interval. | ANSWER: [13, 18]
MCQ
Quick Quiz
Which symbol correctly represents a closed interval where both endpoints are included?
(a, b)
[a, b)
(a, b]
[a, b]
The Correct Answer Is:
D
Option D, [a, b], uses square brackets which mean that both 'a' and 'b' are included in the interval. The other options use round brackets, indicating that one or both endpoints are not included.
Real World Connection
In the Real World
When you book train tickets on IRCTC, you might see a 'Fare Range' for different classes, like [500, 1500] rupees. This means the fare can be anywhere from 500 to 1500 rupees, including both 500 and 1500. Also, in cricket analytics, a player's strike rate might be analyzed within a specific range, say [120, 150], for certain matches.
Key Vocabulary
Key Terms
INTERVAL: A set of real numbers between two given numbers | ENDPOINTS: The two specific numbers that define the start and end of an interval | NUMBER LINE: A visual representation of real numbers as points on a straight line | INCLUDED: Meaning the endpoint value is part of the interval
What's Next
What to Learn Next
Great job learning about closed intervals! Next, you should explore 'Open Intervals' and 'Half-Open/Half-Closed Intervals'. These build on what you've learned and will help you understand even more ways to define ranges of numbers.
