Practice Step Function Intuition in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

Worksheet PDF Free Answer-key PDF Family

Quick 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.

Showing a random 20 of 50 problems.

Example 1

medium
At a jump from value 22 to 55 at x=4x=4, with the left piece including x=4x=4, find f(4)f(4).

Example 2

hard
Define f(x)=⌈x2⌉f(x) = \lceil x^2 \rceil. Find f(1.4)f(1.4) and f(5)f(\sqrt{5}).

Example 3

medium
Let f(x)=⌊x/2⌋f(x) = \lfloor x/2 \rfloor. Find all x∈[0,6)x \in [0, 6) where f(x)=2f(x) = 2.

Example 4

medium
Grades: A for ≥90\ge90, B for ≥80\ge80. A score of 8080 earns what?

Example 5

easy
Evaluate the floor function ⌊3.7⌋\lfloor 3.7 \rfloor.

Example 6

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 7

hard
A parking meter charges $0.50\$0.50 per started 15 minutes. Cost for a 5252-minute park?

Example 8

easy
Does a step function's graph have sloped segments?

Example 9

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

Example 10

medium
If f(x)=⌊x⌋f(x) = \lfloor x \rfloor, compute f(2.999)+f(3)+f(3.001)f(2.999) + f(3) + f(3.001).

Example 11

medium
How many jumps does f(x)=⌊x⌋f(x)=\lfloor x \rfloor have on the interval [0,3)[0,3)?

Example 12

medium
Income tax: 10%10\% on income up to $10,000\$10{,}000, 20%20\% flat on the bracket $10,001\$10{,}001-$30,000\$30{,}000 (whole bracket, not marginal). What does this rate function look like at $15,000\$15{,}000?

Example 13

medium
A step function is $5 for 0≤x<20\le x<2 and $9 for 2≤x<42\le x<4. Find the total jump size at x=2x=2.

Example 14

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

Example 15

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

Example 16

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

Example 17

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

Example 18

challenge
For f(x)=⌊2x⌋f(x)=\lfloor 2x \rfloor, find all jump points in [0,2)[0,2).

Example 19

easy
Evaluate ⌊0⌋\lfloor 0 \rfloor.

Example 20

medium
Is ⌊x⌋+⌈x⌉\lfloor x \rfloor + \lceil x \rceil always 2x2x?
← Back to Step Function Intuition Worked Examples Formula Explained

Related Concepts

Piecewise Function