Piecewise Functions
Functions defined by different rules on different intervals.
Key idea
Functions defined by different rules on different intervals.
In depth
A piecewise function uses different formulas on different parts of its domain. Evaluate by deciding which interval the input lies in. Check edges for open or closed endpoints.
Worked example
f(x) = x² if x < 0, 2x + 1 if x ≥ 0. Then f(−3) = 9, f(3) = 7.
Formula
Real-world uses
- Tax bands, parking charges and postal rates that change at set thresholds
Common mistake
Using the wrong formula at the boundary value.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Piecewise Functions?
Functions defined by different rules on different intervals. It is a topic at the advanced level in Algebra.
What is the formula for Piecewise Functions?
The key formula is abs(x).
Where is Piecewise Functions used in real life?
Real-world examples: Tax bands, parking charges and postal rates that change at set thresholds.
What should I learn before Piecewise Functions?
Make sure you are comfortable with Square Root Functions, the previous topic in Algebra.
What is a common mistake with Piecewise Functions?
Using the wrong formula at the boundary value.
How can I practise Piecewise Functions?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Piecewise Functions?
Move on to Absolute Value Functions next.
Related advanced topics in Algebra
- Absolute Value Inequalities – Rewrite as a compound inequality.
- Systems with Three Variables – Reduce to two variables then solve in sequence.
- Systems of Inequalities – Solution region is the overlap of half-planes.
- Factoring by Grouping – Group terms in pairs to find a common binomial.
- Sum and Difference of Cubes – Special factoring patterns for cubes.
- Completing the Square – Rewrite a quadratic as a perfect square plus constant.