Step Function Intuition Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Step Function Intuition.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A step function is piecewise constant — it takes a fixed value on each of several intervals, jumping abruptly at the interval boundaries.

Imagine a staircase: the height is constant on each step, then jumps up (or down) at each transition. Postal rates, grade cutoffs, and floor() all create steps.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: A step function holds one constant value across a whole interval, then leaps to a new value at the boundary.

Common stuck point: The procedure for step function intuition is the easy part; the trap is interpolating between steps. Asking "Is the output constant within each interval and changing only by sudden jumps at boundaries?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is the output constant within each interval and changing only by sudden jumps at boundaries?

Worked Examples

Example 1

easy

Evaluate the floor function

f(x) = \lfloor x \rfloor

x = 3.7

x = -2.1

, and

x = 5

. Then describe the graph on

[0, 4]

Answer

\lfloor 3.7\rfloor=3

;

\lfloor -2.1\rfloor=-3

;

\lfloor 5\rfloor=5

First step

\lfloor 3.7 \rfloor = 3

(greatest integer

\leq 3.7

Full solution

2
$\lfloor -2.1 \rfloor = -3$ (greatest integer $\leq -2.1$ is $-3$ , not $-2$ ).
3
$\lfloor 5 \rfloor = 5$ . Graph on $[0,4]$ : horizontal steps at heights $0,1,2,3$ . Each step spans a half-open interval $[n, n+1)$ with a closed left endpoint and open right endpoint.

The floor function always rounds down toward

-\infty

. For negative numbers, this means

\lfloor -2.1\rfloor=-3

, not

-2

, because

-3 \leq -2.1 < -2

Example 2

medium

A parking garage charges

\$3

for the first hour (or part thereof) and

\$2

for each additional hour (or part). Write and evaluate the cost function for

t = 0.5

1

1.2

, and

3.9

hours.

Example 3

medium

Let

f(x) = \lfloor x/2 \rfloor

. Find all

x \in [0, 6)

where

f(x) = 2

Example 4

medium

A rideshare charges

\$2.50

for the first 5 minutes (or part) and

\$1

for each started 5-minute block after. Cost for a

13

-minute trip?

Example 5

medium

A movie ticket service charges

\$10

for

0

1

ticket,

\$18

for

2

tickets,

\$24

for

3

, then

\$5

per additional ticket. Cost for

5

tickets?

Example 6

hard

Find all integers

n

[1, 30]

such that

\lfloor n/3 \rfloor = \lfloor n/4 \rfloor

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Evaluate: (a)

\lfloor 7.9 \rfloor

, (b)

\lceil 4.1 \rceil

, (c)

\lfloor -0.5 \rfloor

, (d)

\lceil -3.2 \rceil

Example 2

hard

Define

f(x) = \lfloor 2x \rfloor

. Find all

x

[0, 2]

where

f(x) = 3

, and sketch

f

[0,2]

Example 3

easy

Evaluate the floor function

\lfloor 3.7 \rfloor

Example 4

easy

Evaluate

\lfloor 5 \rfloor

Example 5

easy

A parking garage charges \$4 for any time up to 1 hour, \$8 up to 2 hours. Cost for 1.5 hours?

Example 6

easy

Does a step function's graph have sloped segments?

Example 7

easy

Is a step function continuous at its jump points?

Example 8

easy

Evaluate

\lfloor -1.2 \rfloor

Example 9

easy

Postage is \$1 for up to 1 oz, \$2 for up to 2 oz. Cost for exactly a 1 oz letter (boundary included in first tier)?

Example 10

easy

How many distinct values does a 3-step function take?

Example 11

medium

Define

f(x)=\lfloor x \rfloor

. Find

f(2.9)+f(3.1)

Example 12

medium

A taxi charges \$3 base plus \$2 for each started mile (round up). Cost for 2.3 miles?

Example 13

medium

At a jump from value

2

5

x=4

, with the left piece including

x=4

, find

f(4)

Example 14

medium

Grades: A for

\ge90

, B for

\ge80

. A score of

80

earns what?

Example 15

medium

How many jumps does

f(x)=\lfloor x \rfloor

have on the interval

[0,3)

Example 16

medium

A step function is $5 for

0\le x<2

and $9 for

2\le x<4

. Find the total jump size at

x=2

Example 17

medium

Why can't you draw a step function without lifting your pencil?

Example 18

medium

Evaluate

\lfloor 2.5 \rfloor + \lceil 2.5 \rceil

Example 19

challenge

Shipping: \$5 for the first pound, \$3 for each additional started pound. Cost for a 3.2 lb package?

Example 20

challenge

For

f(x)=\lfloor 2x \rfloor

, find all jump points in

[0,2)

Example 21

challenge

A cell-phone plan: \$30 for up to 2 GB, then \$10 per started GB beyond. Cost for 4.5 GB?

Example 22

medium

How many \$1 stamps does a 3.4 oz letter need if each oz (rounded up) needs one stamp?

Example 23

easy

Evaluate

\lfloor 0 \rfloor

Example 24

easy

Evaluate

\lfloor -3.5 \rfloor

Example 25

easy

Evaluate

\lceil -2.3 \rceil

Example 26

easy

True or false: the graph of a step function consists of horizontal segments.

Example 27

easy

Evaluate

\lfloor 9.9 \rfloor - \lfloor 9.1 \rfloor

Example 28

medium

How many jumps does

f(x) = \lfloor 3x \rfloor

have on

[0, 1)

Example 29

medium

Evaluate

\lfloor \pi \rfloor + \lceil \pi \rceil

Example 30

medium

For

f(x) = \lfloor x \rfloor

, sketch on

[-2, 2)

and list the value on each step.

Example 31

medium

Income tax:

10\%

on income up to

\$10{,}000

20\%

flat on the bracket

\$10{,}001

\$30{,}000

(whole bracket, not marginal). What does this rate function look like at

\$15{,}000

Example 32

medium

A boundary-included-on-left step function jumps from

5

9

x=3

. What is

f(3)

Example 33

medium

f(x) = \lfloor x \rfloor

, compute

f(2.999) + f(3) + f(3.001)

Example 34

hard

Solve

\lfloor x \rfloor = x - 0.4

for real

x

Example 35

hard

Define

f(x) = \lceil x^2 \rceil

. Find

f(1.4)

and

f(\sqrt{5})

Example 36

hard

A parking meter charges

\$0.50

per started 15 minutes. Cost for a

52

-minute park?

Example 37

hard

f(x) = \lfloor x \rfloor

left-continuous, right-continuous, or neither at each integer?

Example 38

hard

A photocopy shop charges

\$0.10

per page for the first

50

pages and

\$0.07

per page after. Cost for

80

pages?

Example 39

challenge

Find

\sum_{k=1}^{100} \lfloor \sqrt{k} \rfloor

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Related Concepts

Piecewise Function

Background Knowledge

These ideas may be useful before you work through the harder examples.

piecewise function