A piecewise function is one function built from different formulas on different parts of its domain. It is useful when a single rule cannot describe a situation over all input values, such as tax brackets, shipping costs, phone plans, or absolute value behavior. On a graph, each rule appears only over its assigned interval, so endpoints and breakpoints matter.
Learning piecewise functions helps students connect algebraic rules, graphs, and real world conditions.
Key Facts
- A piecewise function uses different formulas for different input intervals.
- To evaluate f(x), first choose the rule whose condition contains x, then substitute x into that rule.
- Open circle means the endpoint is not included, such as x < 2 or x > 2.
- Closed circle means the endpoint is included, such as x <= 2 or x >= 2.
- A function is continuous at x = a if f(a) exists, lim x to a of f(x) exists, and lim x to a of f(x) = f(a).
- A step function is a piecewise function that stays constant on each interval, such as f(x) = floor(x).
Vocabulary
- Piecewise function
- A function defined by two or more formulas, with each formula used on a specific part of the domain.
- Breakpoint
- An input value where the formula for a piecewise function changes.
- Domain interval
- A set of input values over which one rule of a piecewise function applies.
- Continuity
- A property of a graph that has no hole, jump, or break at a point or across an interval.
- Step function
- A piecewise function made of horizontal segments that jump between constant output values.
Common Mistakes to Avoid
- Using the wrong rule for the input value. Always check the inequality condition before substituting x into a formula.
- Ignoring open and closed circles on the graph. An open circle means that point is not part of the function, while a closed circle means it is included.
- Assuming the graph must connect at every breakpoint. Piecewise functions can have jumps, holes, or sharp corners depending on the endpoint values.
- Listing overlapping domain pieces without checking them. If two rules apply to the same x value and give different outputs, the relation is not a well-defined function.
Practice Questions
- 1 Let f(x) = 2x + 1 for x < 0, x^2 for 0 <= x <= 3, and 10 - x for x > 3. Find f(-2), f(0), f(3), and f(5).
- 2 Graph g(x) = -1 for x < 2, x - 3 for 2 <= x < 5, and 4 for x >= 5. Mark open and closed circles at x = 2 and x = 5.
- 3 A piecewise graph has a closed point at (1, 3) from the left rule and an open point at (1, 5) from the right rule. Explain whether the function is continuous at x = 1 and why.