Grade 10 Exam > Grade 10 Notes > Integrated Math 2 > Cheatsheet: Piecewise Functions
Cheatsheet: Piecewise Functions
1. Definition and Notation
1.1 Basic Definition
| Term | Definition |
|---|---|
| Piecewise Function | A function defined by different expressions on different intervals of the domain |
| Domain | Set of all input values (x-values) for which the function is defined |
| Range | Set of all output values (y-values) produced by the function |
1.2 Standard Notation
General form:
| Notation | Meaning |
|---|---|
| f(x) = { expression₁, if condition₁ expression₂, if condition₂ expression₃, if condition₃ } |
Each piece applies only when its condition is true; conditions must cover entire domain without overlap |
2. Writing Piecewise Functions
2.1 Domain Intervals
| Interval Type | Notation and Meaning |
|---|---|
| x <> | All values less than a (open on right) |
| x ≤ a | All values less than or equal to a (closed on right) |
| x > a | All values greater than a (open on left) |
| x ≥ a | All values greater than or equal to a (closed on left) |
| a < x=""><> | All values between a and b (open interval) |
| a ≤ x ≤ b | All values between a and b including endpoints (closed interval) |
| a ≤ x <> | All values between a and b, including a but not b |
2.2 Common Forms
- Linear pieces: f(x) = mx + b on specific intervals
- Constant pieces: f(x) = c (horizontal lines)
- Quadratic pieces: f(x) = ax² + bx + c on specific intervals
- Combinations of different function types
3. Evaluating Piecewise Functions
3.1 Step-by-Step Process
- Identify the input value (x-value)
- Determine which condition the input satisfies
- Use the corresponding expression for that piece
- Substitute the input value into the expression
- Simplify to find the output
3.2 Key Points
- Each input corresponds to exactly one piece of the function
- Check inequality symbols carefully (< vs="">
- Endpoint values belong to the piece with ≤ or ≥
- If evaluating at a boundary, determine which piece includes that x-value
4. Graphing Piecewise Functions
4.1 Graphing Steps
- Graph each piece separately on its specified interval
- Check endpoints: use closed dot (•) for included values (≤ or ≥)
- Use open dot (○) for excluded values (< or="">)
- Ensure each x-value has exactly one y-value on the graph
- Label important points (endpoints, intercepts)
4.2 Endpoint Notation
| Symbol | Graph Notation |
|---|---|
| ≤ or ≥ | Closed dot (•) - point is included |
| < or=""> | Open dot (○) - point is not included |
4.3 Graph Characteristics
- Continuous function: no breaks or jumps in the graph
- Discontinuous function: has breaks or jumps where pieces don't connect
- Jump discontinuity: occurs when left and right pieces don't meet at a boundary
- Vertical line test: still applies (each x has only one y)
5. Special Piecewise Functions
5.1 Absolute Value Function
| Function | Piecewise Form |
|---|---|
| f(x) = |x| | f(x) = { x, if x ≥ 0 -x, if x < 0=""> |
| f(x) = |x - h| | f(x) = { x - h, if x ≥ h -(x - h), if x < h=""> |
5.2 Step Functions
| Type | Description |
|---|---|
| Greatest Integer (Floor) | f(x) = ⌊x⌋ rounds down to nearest integer; creates horizontal steps |
| Constant Pieces | Horizontal line segments on different intervals; graph looks like steps |
5.3 Real-World Applications
- Tax brackets: different rates for different income ranges
- Shipping costs: rates change based on weight ranges
- Parking fees: rates change based on time intervals
- Utility billing: tiered pricing structures
6. Domain and Range
6.1 Finding Domain
- Combine all intervals from the conditions
- Check if conditions cover all real numbers or specific intervals
- Express using interval notation or inequality notation
6.2 Finding Range
- Evaluate or graph each piece on its interval
- Determine y-values produced by each piece
- Consider endpoints (included or excluded)
- Combine all possible y-values
- Express using interval notation or set notation
7. Continuity
7.1 Testing Continuity at a Point x = a
| Condition | Requirement |
|---|---|
| Left-hand limit | Evaluate piece valid for x < a="" as="" x="" approaches=""> |
| Right-hand limit | Evaluate piece valid for x > a as x approaches a |
| Function value | f(a) must be defined |
| Continuity | Function is continuous at x = a if left limit = right limit = f(a) |
7.2 Types of Discontinuity
- Jump discontinuity: left and right limits exist but are not equal
- Point discontinuity: limits equal but f(a) undefined or different
- Check all boundary points where pieces meet
8. Solving Equations with Piecewise Functions
8.1 Solving f(x) = k
- Set each piece equal to k separately
- Solve each equation algebraically
- Check if solution falls within the interval for that piece
- Reject solutions outside their interval
- Combine all valid solutions
8.2 Example Process
For f(x) = { 2x + 1, if x < 3;="" x²="" -="" 5,="" if="" x="" ≥="" 3="" },="" solve="" f(x)="">
- Piece 1: 2x + 1 = 7 → x = 3, but 3 is not < 3,="" so="">
- Piece 2: x² - 5 = 7 → x² = 12 → x = ±√12, check x ≥ 3, keep x = √12 only
- Solution: x = √12 or x = 2√3
9. Common Mistakes to Avoid
- Using wrong piece when evaluating at boundary points
- Forgetting to check if algebraic solution is in the correct interval
- Confusing open and closed dots on graphs
- Not covering entire domain (leaving gaps in conditions)
- Overlapping conditions (multiple pieces for same x-value)
- Misreading inequality symbols
- Not simplifying expressions after substitution
10. Key Formulas and Relationships
10.1 Important Reminders
| Concept | Formula or Rule |
|---|---|
| Function definition | Each x-value maps to exactly one y-value |
| Absolute value | |a| = { a, if a ≥ 0; -a, if a < 0=""> |
| Distance from h | |x - h| measures distance from x to h on number line |
| Slope | m = (y₂ - y₁)/(x₂ - x₁) for linear pieces |
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Apr 21, 2026
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