Algebra 2
Other functions (advanced)
Step functions
20 practice questions
0 video lessons
Theory + worked examples
Theory
A step function is constant on each interval and jumps at the boundaries:
- Floor \(\lfloor x\rfloor\): the greatest integer \(\le x\) (round down).
- Ceiling \(\lceil x\rceil\): the least integer \(\ge x\) (round up).
Each step is a horizontal segment, closed on one end and open on the other.
The floor function \(\lfloor x\rfloor\) steps down at each integer.
Floor and ceiling functions.
Floor and ceiling:
\[\lfloor x\rfloor=\text{round down},\qquad \lceil x\rceil=\text{round up}\]
Floor of a negative rounds toward \(-\infty\): \(\lfloor-1.3\rfloor=-2\).
How to use step functions
- Decide floor (down) or ceiling (up).
- Round the input to the correct integer.
- For models, match the rounding to the situation.
- Note the jump points at integers.
Example 1 — Floor values
Evaluate \(\lfloor 2.7\rfloor\) and \(\lfloor -1.3\rfloor\).
Solution
Round down to the nearest integer.
| \(\lfloor 2.7\rfloor\) | \(=\) | \(2\) |
| \(\lfloor -1.3\rfloor\) | \(=\) | \(-2\) |
Example 2 — Ceiling value
Evaluate \(\lceil 2.1\rceil\).
Solution
Round up to the nearest integer.
| \(\lceil 2.1\rceil\) | \(=\) | \(3\) |
Example 3 — Pricing model
Parking costs \(\$2\) per hour or part of an hour. Find the cost for \(2.3\) hours.
Solution
Round up the hours with the ceiling.
| \(\lceil 2.3\rceil\) | \(=\) | \(3\) |
| \(\text{cost}\) | \(=\) | \(3\times\$2=\$6\) |
Example 4 — Where it jumps
At what \(x\) does \(\lfloor x\rfloor\) jump between \(2\) and \(3\)?
Solution
The floor jumps at each integer.
| \(x\) | \(=\) | \(3\) |
Common pitfalls
Floor of a negative goes more negative: \(\lfloor-1.3\rfloor=-2\).
Floor rounds down, ceiling rounds up — not toward zero.
Watch which endpoint each step includes.
Frequently asked questions
What is the floor function?
The greatest integer less than or equal to \(x\).
What is \(\lfloor -1.3\rfloor\)?
\(-2\), since floor rounds toward \(-\infty\).
What is a step function?
A function that is constant on intervals and jumps at boundaries.
What is the difference between floor and ceiling?
Floor rounds down; ceiling rounds up.
More in Other functions (advanced)