Physics and Mathematics
Greatest Integer Function
1. Concept Overview
The Greatest Integer Function helps us convert a real number into the largest integer less than or equal to it.
In simple words, it tells us:
“What is the biggest integer that does not exceed the given number?”
2. Definition of Greatest Integer Function
The greatest integer function of [x], denoted by [ {[}x{]} ], is defined as:
[{[}x{]}][ =] the greatest integer [≤ x]
3. Understanding Through Examples
- [{[}3.7{]} = 3]
- [{[}5{]} = 5]
- [{[}−2.4{]} = −3]
Important observation:
For negative numbers, GIF moves to the left on the number line.
4. Domain and Range
Domain:
All real numbers
[(−∞,∞)]
Range:
All integers
[{…, −3, −2, −1, 0, 1, 2, 3, …}]
5. Graph of Greatest Integer Function
- Graph consists of horizontal line segments
- Each segment is closed on the left and open on the right
- Function is discontinuous at every integer
6. Step Nature of GIF
The greatest integer function is also called a step function because it changes value in steps.
7. Properties of Greatest Integer Function
- [{[}x{]} ≤ x < {[}x{]} + 1]
- [{[}x{]} = x] when [x] is an integer
- Discontinuous at all integers
- Continuous on each interval [(n, n+1)]
- Not one–one
8. Examples with Solutions
Example 1. Find the value of [{[}3.6{]}].
Solution:
The greatest integer less than or equal to 3.6 is 3.
So, [{[}3.6{]} = 3]
Example 2. Find the value of [{[}−1.2{]}].
Solution:
The greatest integer less than or equal to −1.2 is −2.
So, [{[}−1.2{]} = −2]
Example 3. Evaluate [{[}5{]}].
Solution:
If the number itself is an integer, the value remains the same.
So, [{[}5{]} = 5]
Example 4. Find the value of [{[}x{]}] when [2 ≤ x < 3].
Solution:
For all values of x in this interval, the greatest integer ≤ x is 2.
So, [{[}x{]} = 2]
Example 5. Find the value of [{[}−x{]}] when [x = 2.7].
Solution:
First compute [−x = −2.7].
The greatest integer ≤ −2.7 is −3.
So, [{[}−x{]} = −3]
9. Conceptual Questions with Solutions
1. What does the greatest integer function represent?
It represents the largest integer less than or equal to x.
2. What is [{[}4.9{]}]?
The greatest integer ≤ 4.9 is 4.
3. What is [{[}−2.1{]}]?
The greatest integer ≤ −2.1 is −3.
4. Is GIF defined for integers?
Yes, and [{[}x{]} = x] when x is an integer.
5. What is the domain of GIF?
The domain is all real numbers.
6. What is the range of GIF?
The range is all integers.
7. Why is GIF discontinuous?
Because its value jumps at every integer.
8. Is GIF continuous anywhere?
Yes, it is continuous on each open interval [(n, n+1)].
9. Is GIF one–one?
No, many values of x give the same output.
10. Why is GIF called a step function?
Because its graph moves in steps, not smoothly.
11. What type of function is GIF?
It is a many–one function.
12. Where does the graph have closed points?
At the left end of each interval.
13. Where does the graph have open points?
At the right end of each interval.
14. What is [{[}x{]}] if [0 ≤ x < 1]?
The value is 0.
15. Why is GIF important?
It is useful in studying graphs, inequalities, and discontinuity.
10. FAQ / Common Misconceptions
1. [{[}x{]}] gives the nearest integer.
False. It gives the greatest integer less than or equal to x.
2. [{[}−2.4{]} = −2]
False. The correct value is −3.
3. GIF is continuous.
False. It is discontinuous at all integers.
4. Range of GIF is real numbers.
False. Range is only integers.
5. GIF is one–one.
False. It is many–one.
6. [{[}x{]} > x]
False. Always [{[}x{]} ≤ x].
7. GIF graph is slanted.
False. It consists of horizontal steps.
8. [{[}x{]}] changes smoothly.
False. It changes in jumps.
9. GIF is undefined at integers.
False. It is defined everywhere.
10. GIF is not useful.
False. It is very important in mathematics.
11. Practice Questions with Step-by-Step Solutions
Question 1. Find the value of [{[}4.9{]}].
Step-by-Step Solution:
Identify the greatest integer less than or equal to 4.9.
The integer just less than 4.9 is 4.
Conclusion:
[{[}4.9{]} = 4]
Question 2. Find the value of [{[}−3.4{]}].
Step-by-Step Solution:
Move to the left of −3.4 on the number line.
The nearest smaller integer is −4.
Conclusion:
[{[}−3.4{]} = −4]
Question 3. Evaluate [{[}7{]}].
Step-by-Step Solution:
Since 7 is an integer, GIF does not change it.
Conclusion:
[{[}7{]} = 7]
Question 4. Find the value of [{[}x{]}] when [1 ≤ x < 2].
Step-by-Step Solution:
In the interval [1,2), the greatest integer [≤ x] is always 1.
Conclusion:
[{[}x{]} = 1]
Question 5. Find the value of [{[}−x{]}] when [x = 1.3].
Step-by-Step Solution:
Substitute x: [−x = −1.3].
The greatest integer [≤ −1.3] is −2.
Conclusion:
[{[}−x{]} = −2]
Question 6. Solve [{[}x{]}] = 3.
Step-by-Step Solution:
[{[}x{]} = 3] means x lies between 3 and 4.
Include 3 but exclude 4.
Conclusion:
[3 ≤ x < 4]
Question 7. Solve [{[}x{]}] = −2.
Step-by-Step Solution:
[{[}x{]} = −2] means x lies between −2 and −1.
Include −2 but exclude −1.
Conclusion:
[−2 ≤ x < −1]
Question 8. Find the domain of [f(x) = {[}x{]}].
Step-by-Step Solution:
Greatest integer function is defined for all real numbers.
Conclusion:
Domain = [(−∞,∞)]
Question 9. Is the greatest integer function one–one?
Step-by-Step Solution:
Many different values of x give the same output.
Example: [{[}2.1{]} = {[}2.8{]} = 2].
Conclusion:
GIF is not one–one.
Question 10. State the points of discontinuity of [{[}x{]}].
Step-by-Step Solution:
The function jumps at every integer value.
Conclusion:
GIF is discontinuous at all integers.
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