Algebra 2 Other functions (advanced)

Step functions

Q: What is the floor function?

The greatest integer at most x.

Q: What is a step function?

A function constant on intervals that jumps at boundaries.

20 practice questions 0 video lessons Theory + worked examples

Step Functions

California Algebra 2 • Standard F-IF.7b • Other Functions

Step Functions is a topic in Other Functions in the California Common Core State Standards. It is aligned to Standard F-IF.7b, which requires students to graph piecewise-defined functions, including step functions.

A step function is constant on each interval and jumps at the boundaries; the floor rounds down and the ceiling rounds up.

Practice 20 questions

Practice questions

Every question with a fully worked solution.

Start practising

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:

\[\lfloor x\rfloor=\text{round down},\qquad \lceil x\rceil=\text{round up}\]

the floor rounds down to an integer; the ceiling rounds 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.

$\lfloor 2.7\rfloor$	$=$	$2$
$\lfloor -1.3\rfloor$	$=$	$-2$

floor of 2.7 is 2, floor of negative 1.3 is negative 2

Example 2 — Ceiling value

Evaluate $\lceil 2.1\rceil$.

Solution

Round up to the nearest integer.

$\lceil 2.1\rceil$

$=$

$3$

ceiling of 2.1 is 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.

$\lceil 2.3\rceil$	$=$	$3$
$\text{cost}$	$=$	$3\times\$2=\$6$

the cost is 6 dollars

Example 4 — Where it jumps

At what $x$ does $\lfloor x\rfloor$ jump between $2$ and $3$?

Solution

The floor jumps at each integer.

$x$

$=$

$3$

it jumps at x equals 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.

← Previous subtopic

Piecewise functions (advanced)

Next subtopic →

Cube and cube root functions

\(\lfloor 2.7\rfloor\)	\(=\)	\(2\)
\(\lfloor -1.3\rfloor\)	\(=\)	\(-2\)

\(\lceil 2.3\rceil\)	\(=\)	\(3\)
\(\text{cost}\)	\(=\)	\(3\times\$2=\$6\)

Step functions

Step Functions

📖 Theory

How to use step functions

Common pitfalls

Frequently asked questions

Theory