AP PRECALCULUS — PREREQUISITE REVIEW
Piecewise-Defined Functions
Notes — Prerequisite Topic 5
💡 Learning Objectives By the end of this lesson you will be able to:
|
1. What is a Piecewise Function?
Most functions you have seen use a single rule for every input in their domain. A piecewise-defined function uses different rules on different parts of its domain. The rule that applies depends on which piece of the domain the input falls into.
A piecewise function is typically written with a brace that groups several rules, each paired with the portion of the domain where it applies. For example:
f(x) = x + 3 if x < −1
f(x) = x² if −1 ≤ x ≤ 2
f(x) = 6 − x if x > 2
The graph of the three-rule piecewise function above. Note the open circle at x = −1 on the first piece and the closed circles at x = −1 and x = 2 on the middle piece. At x = 2, the middle piece ends closed and the next piece starts open so that every input has exactly one output.
2. Evaluating a Piecewise Function
To evaluate f(a):
- Step 1: check which inequality the number a satisfies.
- Step 2: apply the rule attached to that inequality.
📘 Example — Evaluating Using the function above, find f(−4), f(−1), f(0), and f(2). x = −4 satisfies x < −1, so f(−4) = −4 + 3 = −1. x = −1 satisfies −1 ≤ x ≤ 2, so f(−1) = (−1)² = 1. x = 0 satisfies −1 ≤ x ≤ 2, so f(0) = 0² = 0. x = 2 satisfies −1 ≤ x ≤ 2, so f(2) = 2² = 4. |
3. Graphing a Piecewise Function
Sketching is a piece-by-piece operation.
- Step 1: graph each rule as if it applied everywhere.
- Step 2: erase everything outside the domain restriction for that rule.
- Step 3: place a closed dot at a boundary value that is included, and an open dot at one that is not.
⚠️ Closed vs. open dots at the boundary At any boundary value, one of the pieces will include that value (use a filled dot) and the other, if any, will exclude it (use an open dot). A graph cannot have two filled dots stacked vertically at the same x-value. If that happens, the rule isn’t a function there. |
4. The Absolute Value as a Piecewise Function
The absolute value is the canonical piecewise function. By definition:
|x| = x if x ≥ 0
|x| = −x if x < 0
The graph of y = |x|. To the right of 0 the rule is y = x; to the left of 0 the rule is y = −x. The two rays meet at the origin to form a V-shape.
📘 Example — Rewriting an absolute value Express f(x) = |2x − 6| as a piecewise function. The expression inside changes sign at 2x − 6 = 0, i.e. x = 3. For x ≥ 3, 2x − 6 ≥ 0, so |2x − 6| = 2x − 6. For x < 3, 2x − 6 < 0, so |2x − 6| = −(2x − 6) = 6 − 2x. Thus f(x) = { 2x − 6 if x ≥ 3; 6 − 2x if x < 3 }. |
5. Modeling with Piecewise Functions
Piecewise functions appear naturally whenever the underlying rule changes at specific thresholds.
Situation | Why it's piecewise |
|---|---|
Income tax brackets | A different marginal rate applies in each income band |
Shipping or postage rates | Costs jump at each new weight tier |
Mobile phone plans | A flat rate up to a usage cap, then a per-unit rate |
Parking garage fees | First hour free, different rate per hour after that |
Step function (ceiling/floor) | The output jumps whenever the input crosses an integer |
📘 Example — Modeling a shipping rate A shipper charges $5 for packages of up to 1 kg, $8 for more than 1 kg and up to 5 kg, and $12 for packages above 5 kg. Let C(w) be the cost in dollars for a package of weight w kg. C(w) = 5 if 0 < w ≤ 1 C(w) = 8 if 1 < w ≤ 5 C(w) = 12 if w > 5 The graph of C is a step function: flat segments with jumps at w = 1 and w = 5. |
6. Domain, Range, and Continuity
💡 Reading the domain and range
|
📘 Example — Is it continuous at the seam? For f(x) = { 2x + 1 if x < 1; 3x if x ≥ 1 }, the left piece approaches 2(1) + 1 = 3 as x → 1⁻, and the right piece gives f(1) = 3. The two values agree, so f is continuous at x = 1. By contrast, g(x) = { x + 2 if x < 0; x² if x ≥ 0 } has left limit 2 and g(0) = 0 at the seam — not continuous at x = 0. |
7. Summary
- A piecewise function uses different rules on different parts of its domain
- To evaluate, first locate the input in the correct piece, then apply that piece's rule
- When graphing, include or exclude endpoints with filled/open dots so each x-value has exactly one output
- Absolute value is a piecewise function; recognizing it makes many problems cleaner
- Check continuity at boundary points by comparing the values from the two adjacent pieces