Piecewise-Defined Functions
Complete Notes & Formulae for Eleventh Grade (Algebra 2)
1. What is a Piecewise-Defined Function?
Definition:
A piecewise-defined function is a function that is defined by two or more different expressions (sub-functions), each applying to different parts (intervals) of the domain.
General Notation:
\[ f(x) = \begin{cases} \text{expression}_1, & \text{if } \text{condition}_1 \\ \text{expression}_2, & \text{if } \text{condition}_2 \\ \text{expression}_3, & \text{if } \text{condition}_3 \\ \vdots & \vdots \end{cases} \]
Example: Absolute Value as Piecewise Function
\[ |x| = \begin{cases} -x, & \text{if } x < 0 \\ x, & \text{if } x \geq 0 \end{cases} \]
• For negative values, negate the input (make it positive)
• For non-negative values, keep the input as is
Key Characteristics:
✓ Different formulas apply to different parts of the domain
✓ The subdomains must cover the entire domain (no gaps)
✓ Subdomains should not overlap (usually disjoint)
✓ Each piece is defined on a specific interval
2. Evaluate Piecewise-Defined Functions
Step-by-Step Evaluation Process:
Step 1: Identify the input value (x-value)
• Determine which value you need to evaluate
Step 2: Determine which condition/interval the input belongs to
• Check each condition (inequality) in the piecewise function
• Find which condition is TRUE for your input value
Step 3: Use the corresponding expression
• Select the formula that corresponds to the TRUE condition
Step 4: Substitute and evaluate
• Substitute the input value into the selected expression
• Simplify to get the final answer
Detailed Example 1:
Given:
\[ f(x) = \begin{cases} 4x + 5, & \text{if } x < 2 \\ 3x - 8, & \text{if } x \geq 2 \end{cases} \]
Find: \( f(-2) \), \( f(2) \), and \( f(5) \)
Solution for \( f(-2) \):
• Check: Is -2 < 2? YES ✓
• Use: \( f(x) = 4x + 5 \)
• Substitute: \( f(-2) = 4(-2) + 5 = -8 + 5 = -3 \)
Answer: \( f(-2) = -3 \)
Solution for \( f(2) \):
• Check: Is 2 < 2? NO
• Check: Is 2 ≥ 2? YES ✓
• Use: \( f(x) = 3x - 8 \)
• Substitute: \( f(2) = 3(2) - 8 = 6 - 8 = -2 \)
Answer: \( f(2) = -2 \)
Solution for \( f(5) \):
• Check: Is 5 < 2? NO
• Check: Is 5 ≥ 2? YES ✓
• Use: \( f(x) = 3x - 8 \)
• Substitute: \( f(5) = 3(5) - 8 = 15 - 8 = 7 \)
Answer: \( f(5) = 7 \)
Detailed Example 2:
Given:
\[ g(x) = \begin{cases} x^2, & \text{if } x \leq -1 \\ 2x + 3, & \text{if } -1 < x < 3 \\ -x + 7, & \text{if } x \geq 3 \end{cases} \]
Find: \( g(-3) \), \( g(0) \), and \( g(4) \)
Solution for \( g(-3) \):
• -3 ≤ -1? YES ✓
• Use: \( g(x) = x^2 \)
• \( g(-3) = (-3)^2 = 9 \)
Answer: \( g(-3) = 9 \)
Solution for \( g(0) \):
• -1 < 0 < 3? YES ✓
• Use: \( g(x) = 2x + 3 \)
• \( g(0) = 2(0) + 3 = 3 \)
Answer: \( g(0) = 3 \)
Solution for \( g(4) \):
• 4 ≥ 3? YES ✓
• Use: \( g(x) = -x + 7 \)
• \( g(4) = -4 + 7 = 3 \)
Answer: \( g(4) = 3 \)
Important Tips for Evaluation:
⚠️ Pay close attention to ≤ vs < and ≥ vs >
⚠️ Boundary values (like x = 2 in Example 1) can only belong to ONE piece
⚠️ Check which condition is satisfied before substituting
⚠️ Only use ONE expression per evaluation (never combine pieces)
3. Graph Piecewise-Defined Functions
Step-by-Step Graphing Process:
Step 1: Graph Each Piece Separately
• Treat each sub-function as a regular function
• Graph using appropriate method (table, transformations, etc.)
Step 2: Restrict Each Piece to Its Domain
• Only draw the portion that satisfies the given condition
• Erase or ignore parts outside the specified interval
Step 3: Mark Endpoints Correctly
• Closed dot (●): Use for ≤ or ≥ (inclusive endpoints)
• Open dot (○): Use for < or > (exclusive endpoints)
Step 4: Combine All Pieces on One Graph
• Draw all pieces on the same coordinate plane
• Verify no gaps or overlaps in the domain
Graphing Example:
Graph:
\[ h(x) = \begin{cases} x + 2, & \text{if } x < 1 \\ 3, & \text{if } x = 1 \\ -x + 4, & \text{if } x > 1 \end{cases} \]
Piece 1: \( y = x + 2 \) for \( x < 1 \)
• This is a line with slope 1, y-intercept 2
• Draw only for x < 1 (everything to the LEFT of x = 1)
• At x = 1: \( y = 1 + 2 = 3 \), use open dot (○) at (1, 3)
• Extends with arrow to the left
Piece 2: \( y = 3 \) for \( x = 1 \)
• This is just a single point
• Plot closed dot (●) at (1, 3)
Piece 3: \( y = -x + 4 \) for \( x > 1 \)
• This is a line with slope -1, y-intercept 4
• Draw only for x > 1 (everything to the RIGHT of x = 1)
• At x = 1: \( y = -1 + 4 = 3 \), use open dot (○) at (1, 3)
• Extends with arrow to the right
Final Graph Description:
• Line from left with open circle at (1, 3)
• Closed dot at (1, 3)
• Line to the right with open circle at (1, 3)
• The function IS defined at x = 1 (closed dot takes precedence)
Endpoint Notation Guide:
| Condition | Symbol | Meaning | Graph Notation |
|---|---|---|---|
| \( x < a \) | Less than | Does NOT include endpoint | Open dot (○) |
| \( x \leq a \) | Less than or equal | INCLUDES endpoint | Closed dot (●) |
| \( x > a \) | Greater than | Does NOT include endpoint | Open dot (○) |
| \( x \geq a \) | Greater than or equal | INCLUDES endpoint | Closed dot (●) |
| \( x = a \) | Equal to | Only that exact point | Closed dot (●) |
Common Graph Types in Piecewise Functions:
Linear Pieces:
• \( y = mx + b \) → straight line segments
Constant Pieces:
• \( y = c \) → horizontal line segments
Quadratic Pieces:
• \( y = ax^2 + bx + c \) → parabola segments
Absolute Value Pieces:
• \( y = |x| \) → V-shaped segments
4. Domain and Range of Piecewise Functions
Finding Domain:
Definition:
The domain is the set of all possible input values (x-values) for which the function is defined.
Method:
• Identify all intervals/conditions in the piecewise function
• Take the UNION of all subdomains (combine all intervals)
• Domain = union of all x-values covered by the conditions
Formula:
Domain = subdomain₁ ∪ subdomain₂ ∪ subdomain₃ ∪ ...
Finding Range:
Definition:
The range is the set of all possible output values (y-values) that the function can produce.
Method:
• Find the range of EACH piece over its restricted subdomain
• Consider endpoint values (check if included/excluded)
• Take the UNION of all individual ranges
• Graph helps visualize the range (look at y-coordinates)
Formula:
Range = range₁ ∪ range₂ ∪ range₃ ∪ ...
Example: Finding Domain and Range
Given:
\[ f(x) = \begin{cases} x^2, & \text{if } x < 0 \\ 2, & \text{if } 0 \leq x \leq 3 \\ x - 1, & \text{if } x > 3 \end{cases} \]
Finding Domain:
• Piece 1: \( x < 0 \) → \( (-\infty, 0) \)
• Piece 2: \( 0 \leq x \leq 3 \) → \( [0, 3] \)
• Piece 3: \( x > 3 \) → \( (3, \infty) \)
• Union: \( (-\infty, 0) \cup [0, 3] \cup (3, \infty) \)
Domain: \( (-\infty, \infty) \) or \( \mathbb{R} \) (all real numbers)
Finding Range:
• Piece 1 (\( y = x^2 \) for \( x < 0 \)): As x approaches 0 from left, y approaches 0. As x → -∞, y → ∞
Range₁ = \( (0, \infty) \)
• Piece 2 (\( y = 2 \) for \( 0 \leq x \leq 3 \)): Constant value
Range₂ = \( \{2\} \)
• Piece 3 (\( y = x - 1 \) for \( x > 3 \)): At x = 3, y = 2 (not included). As x → ∞, y → ∞
Range₃ = \( (2, \infty) \)
• Union: \( (0, \infty) \cup \{2\} \cup (2, \infty) \)
Range: \( (0, \infty) \) (all positive real numbers)
5. Special Properties of Piecewise Functions
Continuity:
A piecewise function is continuous at a point if:
• The function is defined at that point
• The left and right pieces meet at the same y-value
• There is NO jump or gap in the graph
A piecewise function is discontinuous if:
• There's a jump (different y-values from left and right)
• There's a hole (point not defined)
• There's a vertical asymptote
Real-World Applications:
• Tax Brackets:
Different tax rates for different income levels
• Shipping Costs:
Pricing based on weight ranges
• Parking Fees:
Different rates for different time durations
• Cellular Plans:
Different charges based on data usage
• Electric Bills:
Tiered pricing for different usage levels
Common Mistakes to Avoid:
❌ Using the wrong piece for boundary values
❌ Forgetting to check which condition is satisfied
❌ Using closed dots when condition has < or >
❌ Drawing the entire function instead of restricting to subdomain
❌ Confusing domain with range
❌ Not considering endpoint values when finding range
6. Quick Reference Summary
Evaluation Checklist:
1. Identify the x-value to evaluate
2. Check which condition is satisfied
3. Use only that piece's expression
4. Substitute and simplify
Graphing Checklist:
1. Graph each piece separately
2. Restrict each to its subdomain
3. Mark endpoints: ● for ≤/≥, ○ for
4. Combine all pieces on one graph
Domain & Range:
• Domain = union of all subdomains
• Range = union of all piece ranges over their subdomains
Notation Reference:
\[ f(x) = \begin{cases} \text{expression}_1, & \text{condition}_1 \\ \text{expression}_2, & \text{condition}_2 \\ \text{expression}_3, & \text{condition}_3 \end{cases} \]
📚 Study Tips
✓ Always identify which condition applies before evaluating
✓ Pay careful attention to < vs ≤ and > vs ≥
✓ Use open dots (○) for strict inequalities, closed dots (●) for inclusive
✓ Graph each piece only over its specified interval
✓ Practice with real-world examples to understand applications