Piecewise Functions Worksheet

This piecewise functions worksheet skips the number-line-only version of the topic and builds every example around a rule that changes once a threshold is crossed: a freelancer's flat fee that switches to hourly billing, a bike rental that gets more expensive the longer you keep it, a paycheck that pays overtime past 40 hours. Piecewise functions describe exactly that kind of pricing, which makes them one of the more naturally financial topics in an algebra sequence.

The worksheet is split into two parts. Part 1 introduces case notation using a single running example, a freelance artist named Tyrese who charges a flat project fee up to four hours and an hourly rate beyond that, then has students read a completed piecewise equation, apply it to new numbers, and identify its graph. It also raises the stakes on Tyrese's own pricing and walks through a second scenario, a bike rental with two rate changes, asking students to fill in the equation themselves and evaluate it at several input values. Part 2 hands off nine independent scenarios, covering in-flight internet, bulk bagel pricing, parking rates, event DJ fees, produce pricing, overtime pay, postage, recruiter commissions, and a health insurance cost-sharing structure, and asks students to write the piecewise function for each one from scratch.

Because Part 2 scenarios are described in plain sentences rather than handed over as formulas, the real skill being tested is translation: pulling the threshold, the base rate, and the rate that kicks in afterward out of a word problem and writing them as cases. Some scenarios add a maximum domain (the internet plan caps at 24 hours, the postage scale caps at 3.5 ounces, the insurance plan caps total payments at $8,000), which pushes students to think about domain restrictions, not just the rule itself.

Below is the worksheet exactly as it appears in the Rapunzl curriculum, followed by teacher notes on how to use it in class.

This activity is from Module 14 of the Rapunzl curriculum, Financial Equations.

Piecewise Functions

This activity teaches students about piecewise functions by examining real-world examples and identifying patterns in how different functions are applied. By the end of this exercise, students will be able to create and apply piecewise functions to solve different problems.

1. Tyrese is a freelance artist who charges a flat fee of $60 for any project that takes 4 hours or less. If a project takes him longer than 4 hours, he charges $15 per hour.
   • a. How much does Tyrese charge for a project that takes him 3 hours?
   • b. How much does Tyrese charge for a project that takes him 8 hours?
2. Which of the following graph accurately represents Tyrese’s pricing? How do you know?

Graph A Graph B

Graph C Graph D

Part 1: Introducing Piecewise Functions

A piecewise function has multiple “pieces” that follow different rules, depending on the domain (x values). We can use case notation to write an equation for piecewise functions. This tells us what rule to apply for each “case” or “piece” of the function. Here is the equation for Tyrese’s project fees:

1. Tyrese’s business is booming, so he decides to increase his prices. Now, he charges $100 for any project that takes five hours or less. He charges $20 per hour for projects that take longer than 5 hours. Write the new equation for his pricing.
2. Marshall is renting a bike for the day. It costs $13 for up to one hour. After one hour, the price increases to $20. After three hours, the price increases again to $50. The maximum time he can rent the bike is 10 hours total.
   • a. Fill in the blanks to write an equation that represents this situation.
   • b. What is f(4)?
   • c. What is f(1)?
   • d. For which x values does f(x) = 20?
   • e. What is the domain of this function?

Part 2: Creating Piecewise Functions

Write a piecewise function for each of the scenarios outlined below.

1. An airline charges for in-flight internet. It costs $7 for 1 hour or less of internet. It costs $19 for more than 1 hour, with a maximum of 24 hours total. Write an equation for the cost function.
2. Bagels are $1 each if you buy 12 or fewer bagels. If you buy more than 12 bagels, they cost $0.75 each.
3. A parking lot charges $3 to park for anytime up to 2 hours. After 2 hours, they charge $1.50 per hour
4. Kaustabh is a DJ who charges $200 for any event under 2 hours. For events 2 hours or longer, he charges $100 per hour.
5. Apples cost $3 per pound if you buy less than 5 pounds. If you buy 5 pounds or more, they cost $1.25 per pound.
6. Sara earns $16 per hour working at Foods Co-op. If she works more than 40 hours a week, she earns time-and-a-half (ie. a 50% pay increase) for those overtime hours.
7. A stamp costs $0.58 for any letter that weighs 1 oz or less. It costs an additional $0.20 per ounce to mail letters that weigh more than 1 oz, up to a maximum of 3.5 oz.
8. Cleo works as a recruiter earning commission. They have a base salary of $35,000 and are paid $5,000 per person hired for the first 8 positions they fill. If they fill more than 8 positions, they are paid $6000 for each additional person hired.
9. Under her health insurance, Toni pays 100% of her medical costs up to $1000. After that, she pays 30% of any additional medical costs. However, her total payments are limited to $8000, no matter her medical costs.

Teacher Notes

Work through the Tyrese example on the board before releasing students to Part 1's second and third questions. The case notation table is the moment students either get comfortable with the ≤ / > boundary language or start guessing, and it's easier to catch confusion live than in a stack of independent answers.

Part 2's nine scenarios use inconsistent threshold language on purpose: "or less," "or fewer," "under," "up to," and "or longer" all show up, and they don't all mean the same thing at the boundary value. That's worth naming explicitly, since a student who treats "5 pounds or more" and "more than 5 pounds" as interchangeable will write a technically wrong piece.

A few scenarios cap the domain outright (the internet plan, the postage scale, the insurance plan), while others are open-ended. Ask students to state the domain for each piece before they write the rule that goes with it. That ordering catches domain errors before they get baked into the equation, rather than after.

Question 2's graph-matching prompt asks students to reason about Tyrese's pricing shape rather than plot it themselves, so it works best paired with whatever graphing tool or handout your class already uses for piecewise graphs.

This worksheet is one piece of the full Financial Equations unit inside the Rapunzl teacher portal, where activities like this one sit alongside articles, guided practice, and a classroom investing simulator built for grades 6–12.

Get a Rapunzl Demo