Piecewise functions use different rules for different parts of the domain. Students need this cheat sheet to read interval conditions, choose the correct formula, and evaluate values accurately. It is especially useful for graphing real-world models such as tax brackets, shipping costs, and step pricing.
The goal is to make each piece of the function clear and organized.
The most important idea is that the input value decides which rule to use. Each condition, such as or , tells where that formula applies. Graphing a piecewise function means graphing each rule only on its assigned interval.
Open and closed circles show whether endpoint values are included.
Key Facts
- A piecewise function is written with multiple rules, such as .
- To evaluate , first find which condition contains , then substitute into only that rule.
- A condition with or uses an open circle at the endpoint because the endpoint is not included.
- A condition with or uses a closed circle at the endpoint because the endpoint is included.
- The domain is the set of all allowed input values covered by the conditions.
- The range is the set of all output values produced by the pieces of the function.
- A piecewise function is continuous at when the left-hand value, right-hand value, and function value are all equal: .
- If two conditions overlap, such as and , the function may be undefined or ambiguous at unless the rules give the same output.
Vocabulary
- Piecewise function
- A function defined by different formulas on different parts of its domain.
- Interval
- A set of input values between given endpoints, such as .
- Endpoint
- A boundary value where one piece of a piecewise function starts or stops.
- Open circle
- A graph symbol showing that an endpoint is not included, usually for or .
- Closed circle
- A graph symbol showing that an endpoint is included, usually for or .
- Continuity
- A property where a graph has no break, jump, or hole at a point.
Common Mistakes to Avoid
- Using every formula to evaluate one input is wrong because only the condition containing that input should be used.
- Ignoring endpoint symbols is wrong because and give different inclusion rules at .
- Graphing each rule across all real numbers is wrong because each formula only applies on its stated interval.
- Choosing the wrong piece for negative numbers is wrong because inequalities must be checked carefully, especially with conditions like and .
- Assuming a piecewise graph is always continuous is wrong because jumps, holes, or mismatched endpoint values can occur.
Practice Questions
- 1 For , find and .
- 2 For , find , , and .
- 3 Graph and label all open and closed circles.
- 4 A piecewise function has for and for . Explain whether the function is continuous at .
Understanding Piecewise Functions Reference
A piecewise rule is really a set of instructions with boundaries. The boundaries divide the input number line into regions. For the relation to be a function, every allowed input must produce one output only.
A gap between conditions means some inputs are not part of the domain. An overlap can cause trouble when the two formulas give different outputs for the same input. If both formulas give the same output at an overlap, the graph may still represent a function, but the notation is clearer when one rule owns the boundary.
When graphing, start by treating each formula as its own familiar graph. A linear rule makes part of a line. A quadratic rule makes part of a parabola.
A constant rule makes a horizontal segment. Then erase every part that lies outside that rule's assigned inputs. This prevents a common mistake of drawing an entire line or curve.
Endpoints deserve extra care because they tell the reader exactly where one instruction stops and another begins. After graphing, find the range by looking vertically at all heights reached by the graph. The range is not always as simple as the listed formulas, since separate pieces may produce the same outputs or leave missing heights.
Continuity describes whether the graph can be drawn through a boundary without lifting a pencil. At a joining input, follow the graph from the left and note the height it approaches. Then follow from the right.
If those heights match, check the actual included point. All three must agree for continuity. A filled point placed away from the joining height creates a removable discontinuity.
A jump occurs when the left and right sides approach different heights. Some piecewise models are meant to jump.
A parking fee can change suddenly after a time limit. Other models should be continuous because the quantity changes smoothly, such as distance traveled over time.
Piecewise functions appear whenever a rule changes after a threshold. Phone plans may charge one rate up to a data limit, then a different rate beyond it. Tax systems use income brackets.
Shipping prices may depend on weight intervals. In these settings, carefully read whether the threshold itself belongs to the lower bracket or upper bracket. When solving problems, make a small number line first and mark each boundary.
Put the test input on that number line before doing any substitution. For graph questions, label endpoints before sketching the pieces.
For continuity questions, calculate each side separately rather than trusting how the graph looks. These habits make complicated-looking definitions much easier to manage.