Greatest Integer Function — Graph, Domain, Examples
Greatest Integer Function — Graph, Domain, Examples
TL;DR
The greatest integer function, written ⌊x⌋, returns the largest integer that is less than or equal to x — so it rounds every number down to a whole number. This article covers its definition, the staircase graph, domain and range, key properties, the rule for negative numbers, and six worked examples.
What Is the Greatest Integer Function?
The greatest integer function takes a real number x and returns the greatest integer that is less than or equal to x. It is written ⌊x⌋ and is also called the floor function or the step function.
In plain terms: slide left along the number line from x until you hit the first integer, and that integer is the answer. For a number that's already an integer, the floor is itself. The key word is down — never round to the nearest integer, always round down.
How Do You Solve the Greatest Integer Function?
The fastest method is the number-line picture: mark x, then take the first integer at or to its left.
- ⌊4.7⌋=4
- ⌊9⌋=9
- ⌊0.3⌋=0
For positive numbers this matches "chop off the decimal." For negative numbers it does not — and that's the catch the next section settles.
Why is the floor of a negative number not what most people expect?
Because "less than or equal to" points the other way for negatives. The greatest integer ≤−2.4 is −3. The rule: for a negative non-integer, the floor goes to the more negative integer.
What Does the Graph of the Greatest Integer Function Look Like?
The graph is a staircase — flat horizontal steps that jump up by one at each integer. Each step carries a solid dot on its left end (the integer is included) and an open dot on its right end (the next integer belongs to the next step).
What Are the Domain and Range of the Greatest Integer Function?
You can feed in any real number, so the domain is all real numbers, R. But the output is always a whole number, so the range is the set of all integers, Z.
What Are the Properties of the Greatest Integer Function?
- Integer shift: ⌊x+n⌋=⌊x⌋+n for any integer n.
- Bounds: ⌊x⌋≤x<⌊x⌋+1 always — the defining inequality.
- Integer inputs: ⌊n⌋=n when n is already an integer.
- Negatives: ⌊−x⌋=−⌈x⌉, linking the floor to its sibling the ceiling function.
- Not one-to-one: many inputs share one output.
Examples of Greatest Integer Function
Example 1
Evaluate ⌊7.92⌋. Final answer: ⌊7.92⌋=7.
Example 2
Evaluate ⌊−3.6⌋. Final answer: ⌊−3.6⌋=−4.
Example 3
Evaluate ⌊5⌋ and ⌊−5⌋. Final answer: ⌊5⌋=5 and ⌊−5⌋=−5.
Example 4
Solve ⌊x⌋=3. Final answer: x∈[3,4).
Example 5
Evaluate ⌊2.5⌋+⌊−2.5⌋. Final answer: −1.
Example 6
A parking meter charges per started hour. How many hours is it billed? Final answer: 3 hours billed.
Where the Greatest Integer Function Earns Its Place
- Computing and programming: integer division, array indexing, etc.
- Billing and tiered pricing: parking, data overages, etc.
- Calendars and clocks.
Where Students Trip Up on the Greatest Integer Function
Mistake 1: Treating the floor of a negative as "chop the decimal".
Mistake 2: Putting the dots on the wrong ends of the graph steps.
Mistake 3: Assuming ⌊x⌋+⌊−x⌋=0.
Key Takeaways
- The greatest integer function ⌊x⌋ returns the largest integer less than or equal to x — it always rounds down.
- For negative non-integers, it moves to the more negative integer.
- The graph is a staircase with solid dots on each step's left end and open dots on its right; the domain is all reals, the range is all integers.
Practice These Before Moving On
- Evaluate ⌊8.99⌋ and ⌊−8.99⌋.
- Solve ⌊x⌋=−2.
- Evaluate ⌊3.5⌋+⌊−3.5⌋.