Practice Step Function Intuition in Math

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

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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

2

5

x=4

, with the left piece including

x=4

, find

f(4)

Example 2

hard

Define

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

. Find

f(1.4)

and

f(\sqrt{5})

Example 3

medium

Let

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

. Find all

x \in [0, 6)

where

f(x) = 2

Example 4

medium

Grades: A for

\ge90

, B for

\ge80

. A score of

80

earns what?

Example 5

easy

Evaluate the floor function

\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

per started 15 minutes. Cost for a

52

-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

f(x) = \lfloor x \rfloor

, compute

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

Example 11

medium

How many jumps does

f(x)=\lfloor x \rfloor

have on the interval

[0,3)

Example 12

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 13

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 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)=\lfloor 2x \rfloor

, find all jump points in

[0,2)

Example 19

easy

Evaluate

\lfloor 0 \rfloor

Example 20

medium

\lfloor x \rfloor + \lceil x \rceil

always

2x

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Related Concepts

Piecewise Function