Piecewise-Defined Functions - Ninth Grade Math
Introduction to Piecewise-Defined Functions
Piecewise-Defined Function: A function that has different rules (formulas) for different parts of its domain
Also Called: Piecewise function, split function, or function defined by cases
Key Characteristic: Different expressions apply depending on the input value
Also Called: Piecewise function, split function, or function defined by cases
Key Characteristic: Different expressions apply depending on the input value
General Notation for Piecewise Functions:
$$f(x) = \begin{cases} \text{expression}_1 & \text{if condition}_1 \\ \text{expression}_2 & \text{if condition}_2 \\ \text{expression}_3 & \text{if condition}_3 \\ \vdots & \vdots \end{cases}$$
Each "piece" has:
• An expression (formula)
• A condition (domain restriction)
$$f(x) = \begin{cases} \text{expression}_1 & \text{if condition}_1 \\ \text{expression}_2 & \text{if condition}_2 \\ \text{expression}_3 & \text{if condition}_3 \\ \vdots & \vdots \end{cases}$$
Each "piece" has:
• An expression (formula)
• A condition (domain restriction)
Why Use Piecewise Functions?
• Model real-world situations with different rules for different conditions
• Describe functions that behave differently in different intervals
• Examples: Tax brackets, shipping costs, parking fees, cell phone plans
Real-World Example:
Taxi fare: $5 base fee + $2 per mile up to 10 miles, then $1.50 per mile after
• Model real-world situations with different rules for different conditions
• Describe functions that behave differently in different intervals
• Examples: Tax brackets, shipping costs, parking fees, cell phone plans
Real-World Example:
Taxi fare: $5 base fee + $2 per mile up to 10 miles, then $1.50 per mile after
Simple Example: Temperature-based discount
$$f(x) = \begin{cases} 10 & \text{if } x \leq 2 \\ 5x & \text{if } x > 2 \end{cases}$$
Interpretation:
• When $x \leq 2$: The function always equals 10
• When $x > 2$: The function equals $5x$
• At $x = 2$: We use the first rule, so $f(2) = 10$
$$f(x) = \begin{cases} 10 & \text{if } x \leq 2 \\ 5x & \text{if } x > 2 \end{cases}$$
Interpretation:
• When $x \leq 2$: The function always equals 10
• When $x > 2$: The function equals $5x$
• At $x = 2$: We use the first rule, so $f(2) = 10$
1. Evaluate Piecewise-Defined Functions
Evaluating a Piecewise Function: Finding the output value for a given input by determining which piece (rule) to use
Steps to Evaluate a Piecewise Function at $x = a$:
Step 1: Look at the given input value $x = a$
Step 2: Check each condition to determine which piece applies
Step 3: Select the correct expression based on the condition that's satisfied
Step 4: Substitute $x = a$ into that expression
Step 5: Simplify to find $f(a)$
Step 1: Look at the given input value $x = a$
Step 2: Check each condition to determine which piece applies
Step 3: Select the correct expression based on the condition that's satisfied
Step 4: Substitute $x = a$ into that expression
Step 5: Simplify to find $f(a)$
Key Points:
• Only ONE piece applies for any given input value
• Pay close attention to $<$, $>$, $\leq$, $\geq$ symbols
• $\leq$ or $\geq$ means the boundary value IS included
• $<$ or $>$ means the boundary value is NOT included
• Always check which interval contains your input value
• Only ONE piece applies for any given input value
• Pay close attention to $<$, $>$, $\leq$, $\geq$ symbols
• $\leq$ or $\geq$ means the boundary value IS included
• $<$ or $>$ means the boundary value is NOT included
• Always check which interval contains your input value
Basic Examples
Example 1: Evaluate the function at $x = -2$, $x = 3$, and $x = 5$
$$f(x) = \begin{cases} x + 4 & \text{if } x < 3 \\ 2x - 1 & \text{if } x \geq 3 \end{cases}$$
Find $f(-2)$:
• Check: Is $-2 < 3$? Yes! ✓
• Use first piece: $f(x) = x + 4$
• $f(-2) = -2 + 4 = 2$
• Answer: $f(-2) = 2$
Find $f(3)$:
• Check: Is $3 < 3$? No
• Check: Is $3 \geq 3$? Yes! ✓
• Use second piece: $f(x) = 2x - 1$
• $f(3) = 2(3) - 1 = 6 - 1 = 5$
• Answer: $f(3) = 5$
Find $f(5)$:
• Check: Is $5 \geq 3$? Yes! ✓
• Use second piece: $f(x) = 2x - 1$
• $f(5) = 2(5) - 1 = 10 - 1 = 9$
• Answer: $f(5) = 9$
$$f(x) = \begin{cases} x + 4 & \text{if } x < 3 \\ 2x - 1 & \text{if } x \geq 3 \end{cases}$$
Find $f(-2)$:
• Check: Is $-2 < 3$? Yes! ✓
• Use first piece: $f(x) = x + 4$
• $f(-2) = -2 + 4 = 2$
• Answer: $f(-2) = 2$
Find $f(3)$:
• Check: Is $3 < 3$? No
• Check: Is $3 \geq 3$? Yes! ✓
• Use second piece: $f(x) = 2x - 1$
• $f(3) = 2(3) - 1 = 6 - 1 = 5$
• Answer: $f(3) = 5$
Find $f(5)$:
• Check: Is $5 \geq 3$? Yes! ✓
• Use second piece: $f(x) = 2x - 1$
• $f(5) = 2(5) - 1 = 10 - 1 = 9$
• Answer: $f(5) = 9$
Example 2: Evaluate at $x = -5$, $x = 0$, and $x = 4$
$$g(x) = \begin{cases} x^2 & \text{if } x \leq 0 \\ 3x + 1 & \text{if } 0 < x < 5 \\ 20 & \text{if } x \geq 5 \end{cases}$$
Find $g(-5)$:
• Is $-5 \leq 0$? Yes! ✓
• Use: $g(x) = x^2$
• $g(-5) = (-5)^2 = 25$
• Answer: $g(-5) = 25$
Find $g(0)$:
• Is $0 \leq 0$? Yes! ✓ (First condition applies)
• Use: $g(x) = x^2$
• $g(0) = 0^2 = 0$
• Answer: $g(0) = 0$
Find $g(4)$:
• Is $4 \leq 0$? No
• Is $0 < 4 < 5$? Yes! ✓
• Use: $g(x) = 3x + 1$
• $g(4) = 3(4) + 1 = 13$
• Answer: $g(4) = 13$
$$g(x) = \begin{cases} x^2 & \text{if } x \leq 0 \\ 3x + 1 & \text{if } 0 < x < 5 \\ 20 & \text{if } x \geq 5 \end{cases}$$
Find $g(-5)$:
• Is $-5 \leq 0$? Yes! ✓
• Use: $g(x) = x^2$
• $g(-5) = (-5)^2 = 25$
• Answer: $g(-5) = 25$
Find $g(0)$:
• Is $0 \leq 0$? Yes! ✓ (First condition applies)
• Use: $g(x) = x^2$
• $g(0) = 0^2 = 0$
• Answer: $g(0) = 0$
Find $g(4)$:
• Is $4 \leq 0$? No
• Is $0 < 4 < 5$? Yes! ✓
• Use: $g(x) = 3x + 1$
• $g(4) = 3(4) + 1 = 13$
• Answer: $g(4) = 13$
Example 3: Three-piece function
$$h(x) = \begin{cases} -x + 2 & \text{if } x < -1 \\ x^2 & \text{if } -1 \leq x \leq 2 \\ 2x + 3 & \text{if } x > 2 \end{cases}$$
Evaluate $h(-3)$, $h(-1)$, $h(2)$, $h(4)$:
$h(-3)$: $-3 < -1$ ✓ → Use first piece
$h(-3) = -(-3) + 2 = 3 + 2 = 5$
$h(-1)$: $-1 \leq -1 \leq 2$ ✓ → Use second piece
$h(-1) = (-1)^2 = 1$
$h(2)$: $-1 \leq 2 \leq 2$ ✓ → Use second piece
$h(2) = (2)^2 = 4$
$h(4)$: $4 > 2$ ✓ → Use third piece
$h(4) = 2(4) + 3 = 11$
$$h(x) = \begin{cases} -x + 2 & \text{if } x < -1 \\ x^2 & \text{if } -1 \leq x \leq 2 \\ 2x + 3 & \text{if } x > 2 \end{cases}$$
Evaluate $h(-3)$, $h(-1)$, $h(2)$, $h(4)$:
$h(-3)$: $-3 < -1$ ✓ → Use first piece
$h(-3) = -(-3) + 2 = 3 + 2 = 5$
$h(-1)$: $-1 \leq -1 \leq 2$ ✓ → Use second piece
$h(-1) = (-1)^2 = 1$
$h(2)$: $-1 \leq 2 \leq 2$ ✓ → Use second piece
$h(2) = (2)^2 = 4$
$h(4)$: $4 > 2$ ✓ → Use third piece
$h(4) = 2(4) + 3 = 11$
Special Functions as Piecewise
Example 4: Absolute Value Function
The absolute value function can be written as piecewise:
$$f(x) = |x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}$$
Evaluate:
• $f(5) = 5$ (since $5 \geq 0$, use $f(x) = x$)
• $f(-3) = -(-3) = 3$ (since $-3 < 0$, use $f(x) = -x$)
• $f(0) = 0$ (since $0 \geq 0$, use $f(x) = x$)
The absolute value function can be written as piecewise:
$$f(x) = |x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}$$
Evaluate:
• $f(5) = 5$ (since $5 \geq 0$, use $f(x) = x$)
• $f(-3) = -(-3) = 3$ (since $-3 < 0$, use $f(x) = -x$)
• $f(0) = 0$ (since $0 \geq 0$, use $f(x) = x$)
Word Problem Applications
Example 5: Cell Phone Plan
A cell phone company charges based on data usage:
$$C(g) = \begin{cases} 25 & \text{if } 0 \leq g \leq 2 \\ 25 + 10(g - 2) & \text{if } g > 2 \end{cases}$$
where $C$ is cost in dollars and $g$ is gigabytes used.
Find the cost for 1.5 GB and 5 GB:
$C(1.5)$:
• Is $0 \leq 1.5 \leq 2$? Yes! ✓
• $C(1.5) = 25$
• Cost: $25
$C(5)$:
• Is $5 > 2$? Yes! ✓
• $C(5) = 25 + 10(5 - 2) = 25 + 10(3) = 25 + 30 = 55$
• Cost: $55
A cell phone company charges based on data usage:
$$C(g) = \begin{cases} 25 & \text{if } 0 \leq g \leq 2 \\ 25 + 10(g - 2) & \text{if } g > 2 \end{cases}$$
where $C$ is cost in dollars and $g$ is gigabytes used.
Find the cost for 1.5 GB and 5 GB:
$C(1.5)$:
• Is $0 \leq 1.5 \leq 2$? Yes! ✓
• $C(1.5) = 25$
• Cost: $25
$C(5)$:
• Is $5 > 2$? Yes! ✓
• $C(5) = 25 + 10(5 - 2) = 25 + 10(3) = 25 + 30 = 55$
• Cost: $55
Common Mistakes to Avoid:
• Using the wrong piece (check conditions carefully!)
• Confusing $\leq$ with $<$ (boundary inclusion matters)
• Forgetting to simplify after substitution
• Not checking which condition is satisfied
• Assuming continuity at boundaries (not always true)
• Using the wrong piece (check conditions carefully!)
• Confusing $\leq$ with $<$ (boundary inclusion matters)
• Forgetting to simplify after substitution
• Not checking which condition is satisfied
• Assuming continuity at boundaries (not always true)
2. Graph Piecewise-Defined Functions
Graphing Piecewise Functions: Creating a visual representation by plotting each piece on its specified domain
Open Circle (○): Point NOT included in the graph (use with $<$ or $>$)
Closed Circle (●): Point IS included in the graph (use with $\leq$ or $\geq$)
Open Circle (○): Point NOT included in the graph (use with $<$ or $>$)
Closed Circle (●): Point IS included in the graph (use with $\leq$ or $\geq$)
Steps to Graph a Piecewise Function:
Step 1: Identify all pieces and their domain restrictions
Step 2: For each piece:
a) Graph the function WITHOUT restrictions (lightly)
b) Identify the domain interval for this piece
c) ERASE or darken only the part within the domain
Step 3: Mark endpoints with open or closed circles
• Closed (●) for $\leq$ or $\geq$
• Open (○) for $<$ or $>$
Step 4: Check for continuity at boundary points
Step 5: Label important points
Step 1: Identify all pieces and their domain restrictions
Step 2: For each piece:
a) Graph the function WITHOUT restrictions (lightly)
b) Identify the domain interval for this piece
c) ERASE or darken only the part within the domain
Step 3: Mark endpoints with open or closed circles
• Closed (●) for $\leq$ or $\geq$
• Open (○) for $<$ or $>$
Step 4: Check for continuity at boundary points
Step 5: Label important points
Important Graphing Notes:
• Each piece is graphed ONLY on its specified domain
• Pieces may or may not connect at boundaries
• A function is continuous if all pieces connect without jumps
• A function is discontinuous if there are breaks or jumps
• At boundary points, only ONE piece can have a closed circle
• Each piece is graphed ONLY on its specified domain
• Pieces may or may not connect at boundaries
• A function is continuous if all pieces connect without jumps
• A function is discontinuous if there are breaks or jumps
• At boundary points, only ONE piece can have a closed circle
Graphing Linear Pieces
Example 1: Graph the function
$$f(x) = \begin{cases} x + 2 & \text{if } x < 1 \\ -2x + 5 & \text{if } x \geq 1 \end{cases}$$
Step-by-Step:
Piece 1: $f(x) = x + 2$ for $x < 1$
• This is a line with slope 1, y-intercept 2
• Domain: $x < 1$ (all values less than 1)
• At boundary $x = 1$: $f(1) = 1 + 2 = 3$ (but NOT included)
• Plot: Line from left, ending at $(1, 3)$ with OPEN circle
Piece 2: $f(x) = -2x + 5$ for $x \geq 1$
• This is a line with slope -2, y-intercept 5
• Domain: $x \geq 1$ (1 and all values greater)
• At boundary $x = 1$: $f(1) = -2(1) + 5 = 3$ (IS included)
• Plot: Line from $(1, 3)$ with CLOSED circle, extending right
Key Points:
• At $x = 1$: Open circle from piece 1, closed circle from piece 2
• Both pieces meet at point $(1, 3)$ → Function IS continuous
• Graph has no jumps or breaks
$$f(x) = \begin{cases} x + 2 & \text{if } x < 1 \\ -2x + 5 & \text{if } x \geq 1 \end{cases}$$
Step-by-Step:
Piece 1: $f(x) = x + 2$ for $x < 1$
• This is a line with slope 1, y-intercept 2
• Domain: $x < 1$ (all values less than 1)
• At boundary $x = 1$: $f(1) = 1 + 2 = 3$ (but NOT included)
• Plot: Line from left, ending at $(1, 3)$ with OPEN circle
Piece 2: $f(x) = -2x + 5$ for $x \geq 1$
• This is a line with slope -2, y-intercept 5
• Domain: $x \geq 1$ (1 and all values greater)
• At boundary $x = 1$: $f(1) = -2(1) + 5 = 3$ (IS included)
• Plot: Line from $(1, 3)$ with CLOSED circle, extending right
Key Points:
• At $x = 1$: Open circle from piece 1, closed circle from piece 2
• Both pieces meet at point $(1, 3)$ → Function IS continuous
• Graph has no jumps or breaks
Example 2: Graph with discontinuity
$$g(x) = \begin{cases} 2x - 1 & \text{if } x \leq 2 \\ x + 3 & \text{if } x > 2 \end{cases}$$
Piece 1: $g(x) = 2x - 1$ for $x \leq 2$
• At $x = 2$: $g(2) = 2(2) - 1 = 3$ with CLOSED circle at $(2, 3)$
• Plot line ending at $(2, 3)$ ●
Piece 2: $g(x) = x + 3$ for $x > 2$
• At $x = 2$: $g(2)$ would be $2 + 3 = 5$ but NOT included
• Start with OPEN circle at $(2, 5)$ ○
Analysis:
• At $x = 2$: Closed circle at $(2, 3)$, open circle at $(2, 5)$
• There IS a jump/discontinuity at $x = 2$
• Function value at $x = 2$ is 3 (from first piece)
$$g(x) = \begin{cases} 2x - 1 & \text{if } x \leq 2 \\ x + 3 & \text{if } x > 2 \end{cases}$$
Piece 1: $g(x) = 2x - 1$ for $x \leq 2$
• At $x = 2$: $g(2) = 2(2) - 1 = 3$ with CLOSED circle at $(2, 3)$
• Plot line ending at $(2, 3)$ ●
Piece 2: $g(x) = x + 3$ for $x > 2$
• At $x = 2$: $g(2)$ would be $2 + 3 = 5$ but NOT included
• Start with OPEN circle at $(2, 5)$ ○
Analysis:
• At $x = 2$: Closed circle at $(2, 3)$, open circle at $(2, 5)$
• There IS a jump/discontinuity at $x = 2$
• Function value at $x = 2$ is 3 (from first piece)
Graphing with Constant Pieces
Example 3: Graph with horizontal segments
$$h(x) = \begin{cases} 3 & \text{if } x < -1 \\ 1 & \text{if } -1 \leq x < 2 \\ -2 & \text{if } x \geq 2 \end{cases}$$
Piece 1: $h(x) = 3$ for $x < -1$
• Horizontal line at $y = 3$
• Extends left from $x = -1$ (open circle at $(-1, 3)$)
Piece 2: $h(x) = 1$ for $-1 \leq x < 2$
• Horizontal line segment at $y = 1$
• From $x = -1$ (closed ●) to $x = 2$ (open ○)
• Points: $(-1, 1)$ ● to $(2, 1)$ ○
Piece 3: $h(x) = -2$ for $x \geq 2$
• Horizontal line at $y = -2$
• Starts at $x = 2$ (closed circle at $(2, -2)$), extends right
Result: Step function with jumps at $x = -1$ and $x = 2$
$$h(x) = \begin{cases} 3 & \text{if } x < -1 \\ 1 & \text{if } -1 \leq x < 2 \\ -2 & \text{if } x \geq 2 \end{cases}$$
Piece 1: $h(x) = 3$ for $x < -1$
• Horizontal line at $y = 3$
• Extends left from $x = -1$ (open circle at $(-1, 3)$)
Piece 2: $h(x) = 1$ for $-1 \leq x < 2$
• Horizontal line segment at $y = 1$
• From $x = -1$ (closed ●) to $x = 2$ (open ○)
• Points: $(-1, 1)$ ● to $(2, 1)$ ○
Piece 3: $h(x) = -2$ for $x \geq 2$
• Horizontal line at $y = -2$
• Starts at $x = 2$ (closed circle at $(2, -2)$), extends right
Result: Step function with jumps at $x = -1$ and $x = 2$
Graphing Non-Linear Pieces
Example 4: Graph with quadratic piece
$$f(x) = \begin{cases} x^2 & \text{if } x \leq 1 \\ 2x - 1 & \text{if } x > 1 \end{cases}$$
Piece 1: $f(x) = x^2$ for $x \leq 1$
• Parabola opening upward
• Domain: all x up to and including 1
• Create table of values:
• Plot these points and connect with smooth curve
• Closed circle at $(1, 1)$ ●
Piece 2: $f(x) = 2x - 1$ for $x > 1$
• Linear function with slope 2
• At $x = 1$: would be $2(1) - 1 = 1$ (not included)
• Open circle at $(1, 1)$ ○
• Draw line extending from $(1, 1)$ to the right
Note: Both pieces approach $(1, 1)$, so function IS continuous
$$f(x) = \begin{cases} x^2 & \text{if } x \leq 1 \\ 2x - 1 & \text{if } x > 1 \end{cases}$$
Piece 1: $f(x) = x^2$ for $x \leq 1$
• Parabola opening upward
• Domain: all x up to and including 1
• Create table of values:
| x | -2 | -1 | 0 | 1 |
|---|---|---|---|---|
| y | 4 | 1 | 0 | 1 |
• Closed circle at $(1, 1)$ ●
Piece 2: $f(x) = 2x - 1$ for $x > 1$
• Linear function with slope 2
• At $x = 1$: would be $2(1) - 1 = 1$ (not included)
• Open circle at $(1, 1)$ ○
• Draw line extending from $(1, 1)$ to the right
Note: Both pieces approach $(1, 1)$, so function IS continuous
Absolute Value as Piecewise
Example 5: Graph $f(x) = |x|$ as piecewise
$$f(x) = \begin{cases} -x & \text{if } x < 0 \\ x & \text{if } x \geq 0 \end{cases}$$
Piece 1: $f(x) = -x$ for $x < 0$
• Line with slope -1 (negative x values)
• Points: $(-3, 3)$, $(-2, 2)$, $(-1, 1)$
• Open circle at $(0, 0)$ approaching from left
Piece 2: $f(x) = x$ for $x \geq 0$
• Line with slope 1 (positive x values)
• Closed circle at $(0, 0)$ ●
• Points: $(0, 0)$, $(1, 1)$, $(2, 2)$, $(3, 3)$
Result: Classic V-shaped graph, continuous at $x = 0$
$$f(x) = \begin{cases} -x & \text{if } x < 0 \\ x & \text{if } x \geq 0 \end{cases}$$
Piece 1: $f(x) = -x$ for $x < 0$
• Line with slope -1 (negative x values)
• Points: $(-3, 3)$, $(-2, 2)$, $(-1, 1)$
• Open circle at $(0, 0)$ approaching from left
Piece 2: $f(x) = x$ for $x \geq 0$
• Line with slope 1 (positive x values)
• Closed circle at $(0, 0)$ ●
• Points: $(0, 0)$, $(1, 1)$, $(2, 2)$, $(3, 3)$
Result: Classic V-shaped graph, continuous at $x = 0$
Checking Continuity
A piecewise function is CONTINUOUS at $x = a$ if:
1. Both pieces approach the same y-value at $x = a$
2. $f(a)$ is defined (has a value)
3. The limit from left = limit from right = $f(a)$
Simple Check:
• Evaluate the point from both pieces
• If values match AND point is included → continuous
• If values don't match OR neither included → discontinuous
1. Both pieces approach the same y-value at $x = a$
2. $f(a)$ is defined (has a value)
3. The limit from left = limit from right = $f(a)$
Simple Check:
• Evaluate the point from both pieces
• If values match AND point is included → continuous
• If values don't match OR neither included → discontinuous
Example 6: Check continuity
$$f(x) = \begin{cases} x + 1 & \text{if } x < 2 \\ 2x - 1 & \text{if } x \geq 2 \end{cases}$$
At $x = 2$:
• From left (using first piece): $\lim_{x \to 2^-} (x + 1) = 2 + 1 = 3$
• From right (using second piece): $f(2) = 2(2) - 1 = 3$
• Both equal 3, and $f(2) = 3$ is defined
• Function IS continuous at $x = 2$
$$f(x) = \begin{cases} x + 1 & \text{if } x < 2 \\ 2x - 1 & \text{if } x \geq 2 \end{cases}$$
At $x = 2$:
• From left (using first piece): $\lim_{x \to 2^-} (x + 1) = 2 + 1 = 3$
• From right (using second piece): $f(2) = 2(2) - 1 = 3$
• Both equal 3, and $f(2) = 3$ is defined
• Function IS continuous at $x = 2$
Real-World Applications of Piecewise Functions
Example 1: Parking Garage Fees
A parking garage charges:
• $5 for up to 2 hours
• $5 plus $3 per hour after 2 hours
$$C(h) = \begin{cases} 5 & \text{if } 0 < h \leq 2 \\ 5 + 3(h - 2) & \text{if } h > 2 \end{cases}$$
Find cost for 1.5, 2, and 5 hours:
• $C(1.5) = 5$ dollars
• $C(2) = 5$ dollars
• $C(5) = 5 + 3(5-2) = 5 + 9 = 14$ dollars
A parking garage charges:
• $5 for up to 2 hours
• $5 plus $3 per hour after 2 hours
$$C(h) = \begin{cases} 5 & \text{if } 0 < h \leq 2 \\ 5 + 3(h - 2) & \text{if } h > 2 \end{cases}$$
Find cost for 1.5, 2, and 5 hours:
• $C(1.5) = 5$ dollars
• $C(2) = 5$ dollars
• $C(5) = 5 + 3(5-2) = 5 + 9 = 14$ dollars
Example 2: Tax Brackets (Simplified)
Income tax rate:
• 10% on income up to $10,000
• 15% on income over $10,000
$$T(x) = \begin{cases} 0.10x & \text{if } 0 \leq x \leq 10000 \\ 1000 + 0.15(x - 10000) & \text{if } x > 10000 \end{cases}$$
where $x$ is income and $T(x)$ is tax owed
Income tax rate:
• 10% on income up to $10,000
• 15% on income over $10,000
$$T(x) = \begin{cases} 0.10x & \text{if } 0 \leq x \leq 10000 \\ 1000 + 0.15(x - 10000) & \text{if } x > 10000 \end{cases}$$
where $x$ is income and $T(x)$ is tax owed
Example 3: Shipping Costs
$$S(w) = \begin{cases} 5 & \text{if } 0 < w \leq 1 \\ 8 & \text{if } 1 < w \leq 5 \\ 8 + 2(w - 5) & \text{if } w > 5 \end{cases}$$
where $w$ is weight in pounds, $S(w)$ is shipping cost in dollars
$$S(w) = \begin{cases} 5 & \text{if } 0 < w \leq 1 \\ 8 & \text{if } 1 < w \leq 5 \\ 8 + 2(w - 5) & \text{if } w > 5 \end{cases}$$
where $w$ is weight in pounds, $S(w)$ is shipping cost in dollars
Quick Reference Guide
Piecewise Function Notation:
$$f(x) = \begin{cases} \text{formula}_1 & \text{if condition}_1 \\ \text{formula}_2 & \text{if condition}_2 \\ \text{formula}_3 & \text{if condition}_3 \end{cases}$$
$$f(x) = \begin{cases} \text{formula}_1 & \text{if condition}_1 \\ \text{formula}_2 & \text{if condition}_2 \\ \text{formula}_3 & \text{if condition}_3 \end{cases}$$
Evaluation Process:
1. Identify which condition is satisfied
2. Use the corresponding formula
3. Substitute and simplify
1. Identify which condition is satisfied
2. Use the corresponding formula
3. Substitute and simplify
Graphing Symbols:
• Closed circle (●): Point IS included ($\leq$ or $\geq$)
• Open circle (○): Point is NOT included ($<$ or $>$)
• Only one circle can be closed at any boundary point
• Closed circle (●): Point IS included ($\leq$ or $\geq$)
• Open circle (○): Point is NOT included ($<$ or $>$)
• Only one circle can be closed at any boundary point
| Condition Type | Symbol | Boundary Point | Graph Marking |
|---|---|---|---|
| Less than | $x < a$ | NOT included | Open circle (○) at $x = a$ |
| Less than or equal | $x \leq a$ | IS included | Closed circle (●) at $x = a$ |
| Greater than | $x > a$ | NOT included | Open circle (○) at $x = a$ |
| Greater than or equal | $x \geq a$ | IS included | Closed circle (●) at $x = a$ |
Continuity Check:
At boundary point $x = a$:
• Evaluate left piece approaching $a$
• Evaluate right piece approaching $a$
• If values match AND point included → Continuous
• If values differ → Discontinuous (jump)
At boundary point $x = a$:
• Evaluate left piece approaching $a$
• Evaluate right piece approaching $a$
• If values match AND point included → Continuous
• If values differ → Discontinuous (jump)
Common Examples of Piecewise Functions
Famous Piecewise Functions:
1. Absolute Value: $f(x) = |x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}$
2. Greatest Integer (Floor): $f(x) = \lfloor x \rfloor$ = largest integer $\leq x$
3. Sign Function: $\text{sgn}(x) = \begin{cases} 1 & \text{if } x > 0 \\ 0 & \text{if } x = 0 \\ -1 & \text{if } x < 0 \end{cases}$
4. Unit Step Function: $H(x) = \begin{cases} 0 & \text{if } x < 0 \\ 1 & \text{if } x \geq 0 \end{cases}$
1. Absolute Value: $f(x) = |x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}$
2. Greatest Integer (Floor): $f(x) = \lfloor x \rfloor$ = largest integer $\leq x$
3. Sign Function: $\text{sgn}(x) = \begin{cases} 1 & \text{if } x > 0 \\ 0 & \text{if } x = 0 \\ -1 & \text{if } x < 0 \end{cases}$
4. Unit Step Function: $H(x) = \begin{cases} 0 & \text{if } x < 0 \\ 1 & \text{if } x \geq 0 \end{cases}$
Success Tips for Piecewise Functions:
✓ Always check which condition applies before evaluating
✓ Pay attention to $<$, $>$, $\leq$, $\geq$ symbols
✓ Use closed circles for $\leq$ or $\geq$, open for $<$ or $>$
✓ Graph each piece only on its specified domain
✓ Check continuity at boundary points
✓ Make a table of values for curved pieces
✓ Label all important points and circles clearly
✓ Verify your graph matches the function definition
✓ For word problems, identify the different "cases" or conditions
✓ Practice with real-world examples to understand applications
✓ Always check which condition applies before evaluating
✓ Pay attention to $<$, $>$, $\leq$, $\geq$ symbols
✓ Use closed circles for $\leq$ or $\geq$, open for $<$ or $>$
✓ Graph each piece only on its specified domain
✓ Check continuity at boundary points
✓ Make a table of values for curved pieces
✓ Label all important points and circles clearly
✓ Verify your graph matches the function definition
✓ For word problems, identify the different "cases" or conditions
✓ Practice with real-world examples to understand applications