Function Concepts - Ninth Grade Math
Introduction to Relations and Functions
Relation: A set of ordered pairs $(x, y)$ showing a connection between inputs and outputs
Function: A special relation where each input has EXACTLY ONE output
Domain: The set of all input values (x-values)
Range: The set of all output values (y-values)
Mapping: A visual way to show how inputs relate to outputs
Function: A special relation where each input has EXACTLY ONE output
Domain: The set of all input values (x-values)
Range: The set of all output values (y-values)
Mapping: A visual way to show how inputs relate to outputs
Key Difference:
• All functions are relations ✓
• NOT all relations are functions ✗
Function Rule: Each input can have only ONE output
• Input 3 → Output 5 ✓
• Input 3 → Output 5 AND 7 ✗ (Not a function!)
• All functions are relations ✓
• NOT all relations are functions ✗
Function Rule: Each input can have only ONE output
• Input 3 → Output 5 ✓
• Input 3 → Output 5 AND 7 ✗ (Not a function!)
1. Relations: Convert Between Representations
Four Ways to Represent Relations:
1. Table: Organized rows and columns
2. Graph: Points plotted on coordinate plane
3. Mapping Diagram: Arrows showing connections
4. Set of Ordered Pairs: List like $\{(1, 2), (3, 4)\}$
1. Table: Organized rows and columns
2. Graph: Points plotted on coordinate plane
3. Mapping Diagram: Arrows showing connections
4. Set of Ordered Pairs: List like $\{(1, 2), (3, 4)\}$
Example: Same relation, four ways
1. Set of Ordered Pairs: $\{(1, 3), (2, 5), (4, 9)\}$
2. Table:
3. Graph: Plot points $(1, 3)$, $(2, 5)$, $(4, 9)$ on coordinate plane
4. Mapping:
Domain: {1, 2, 4} → Range: {3, 5, 9}
1 → 3
2 → 5
4 → 9
1. Set of Ordered Pairs: $\{(1, 3), (2, 5), (4, 9)\}$
2. Table:
| x | 1 | 2 | 4 |
|---|---|---|---|
| y | 3 | 5 | 9 |
3. Graph: Plot points $(1, 3)$, $(2, 5)$, $(4, 9)$ on coordinate plane
4. Mapping:
Domain: {1, 2, 4} → Range: {3, 5, 9}
1 → 3
2 → 5
4 → 9
Conversion Tips:
• Table → Ordered Pairs: Each row/column pair is $(x, y)$
• Graph → Ordered Pairs: Read coordinates of each point
• Mapping → Ordered Pairs: Follow each arrow
• Ordered Pairs → Any: Use pairs to create other forms
• Table → Ordered Pairs: Each row/column pair is $(x, y)$
• Graph → Ordered Pairs: Read coordinates of each point
• Mapping → Ordered Pairs: Follow each arrow
• Ordered Pairs → Any: Use pairs to create other forms
2. Domain and Range of Relations
Definitions:
Domain: Set of all input values (x-values)
• All possible x-coordinates
• Written in set notation: $\{x_1, x_2, x_3, ...\}$
Range: Set of all output values (y-values)
• All possible y-coordinates
• Written in set notation: $\{y_1, y_2, y_3, ...\}$
Domain: Set of all input values (x-values)
• All possible x-coordinates
• Written in set notation: $\{x_1, x_2, x_3, ...\}$
Range: Set of all output values (y-values)
• All possible y-coordinates
• Written in set notation: $\{y_1, y_2, y_3, ...\}$
How to Find Domain and Range:
From Ordered Pairs:
• Domain: List all x-values (no repeats)
• Range: List all y-values (no repeats)
From Table:
• Domain: All values in x-row/column
• Range: All values in y-row/column
From Graph:
• Domain: All x-coordinates of points
• Range: All y-coordinates of points
From Mapping:
• Domain: Left side (inputs)
• Range: Right side (outputs)
From Ordered Pairs:
• Domain: List all x-values (no repeats)
• Range: List all y-values (no repeats)
From Table:
• Domain: All values in x-row/column
• Range: All values in y-row/column
From Graph:
• Domain: All x-coordinates of points
• Range: All y-coordinates of points
From Mapping:
• Domain: Left side (inputs)
• Range: Right side (outputs)
Example 1: $\{(2, 5), (3, 7), (4, 5), (6, 8)\}$
Domain: $\{2, 3, 4, 6\}$
Range: $\{5, 7, 8\}$ (Note: 5 appears twice but listed once)
Domain: $\{2, 3, 4, 6\}$
Range: $\{5, 7, 8\}$ (Note: 5 appears twice but listed once)
Example 2: Table
Domain: $\{-1, 0, 1, 2\}$
Range: $\{1, 2, 3, 4\}$
| x | -1 | 0 | 1 | 2 |
|---|---|---|---|---|
| y | 4 | 3 | 2 | 1 |
Range: $\{1, 2, 3, 4\}$
Important Notes:
• List domain and range in order (usually ascending)
• No repeated values in domain or range
• Use curly braces { } for sets
• Domain and range can have different number of elements
• List domain and range in order (usually ascending)
• No repeated values in domain or range
• Use curly braces { } for sets
• Domain and range can have different number of elements
3. Identify Independent and Dependent Variables
Independent Variable: The input variable that is controlled or changed
• Usually represented by x
• The CAUSE in a cause-effect relationship
• Plotted on the horizontal (x) axis
Dependent Variable: The output variable that depends on the input
• Usually represented by y
• The EFFECT in a cause-effect relationship
• Plotted on the vertical (y) axis
• Usually represented by x
• The CAUSE in a cause-effect relationship
• Plotted on the horizontal (x) axis
Dependent Variable: The output variable that depends on the input
• Usually represented by y
• The EFFECT in a cause-effect relationship
• Plotted on the vertical (y) axis
How to Identify:
Ask: "What affects what?"
• Independent variable → affects → Dependent variable
• x-value → determines → y-value
• Input → produces → Output
Ask: "What affects what?"
• Independent variable → affects → Dependent variable
• x-value → determines → y-value
• Input → produces → Output
Example 1: Hours studied vs. test score
Independent: Hours studied (you control this)
Dependent: Test score (depends on hours studied)
Reason: Study time affects test score
Independent: Hours studied (you control this)
Dependent: Test score (depends on hours studied)
Reason: Study time affects test score
Example 2: Number of workers vs. time to complete job
Independent: Number of workers
Dependent: Time to complete
Reason: More workers affects completion time
Independent: Number of workers
Dependent: Time to complete
Reason: More workers affects completion time
Example 3: Temperature vs. ice cream sales
Independent: Temperature
Dependent: Ice cream sales
Reason: Temperature influences how much ice cream is sold
Independent: Temperature
Dependent: Ice cream sales
Reason: Temperature influences how much ice cream is sold
Example 4: Age vs. height
Independent: Age
Dependent: Height
Reason: As age increases, height changes
Independent: Age
Dependent: Height
Reason: As age increases, height changes
4-5. Identify Functions & Vertical Line Test
Function Definition:
A relation where each input (x-value) has EXACTLY ONE output (y-value)
Notation: $f(x)$ read as "f of x"
$f(x) = y$
A relation where each input (x-value) has EXACTLY ONE output (y-value)
Notation: $f(x)$ read as "f of x"
$f(x) = y$
Is It a Function? Test Methods
Method 1: Check Ordered Pairs or Table
• Look at all x-values
• If any x-value appears MORE THAN ONCE with DIFFERENT y-values → NOT a function
• If each x-value appears only once (or with same y-value) → IS a function
• Look at all x-values
• If any x-value appears MORE THAN ONCE with DIFFERENT y-values → NOT a function
• If each x-value appears only once (or with same y-value) → IS a function
Example 1: $\{(1, 3), (2, 5), (3, 7), (4, 9)\}$
All x-values are unique: 1, 2, 3, 4
Answer: YES, this IS a function ✓
All x-values are unique: 1, 2, 3, 4
Answer: YES, this IS a function ✓
Example 2: $\{(1, 3), (2, 5), (1, 7), (4, 9)\}$
x = 1 appears twice with DIFFERENT y-values (3 and 7)
Answer: NO, this is NOT a function ✗
x = 1 appears twice with DIFFERENT y-values (3 and 7)
Answer: NO, this is NOT a function ✗
Example 3: $\{(2, 5), (3, 5), (4, 5)\}$
All x-values are unique: 2, 3, 4
(Same y-value is okay!)
Answer: YES, this IS a function ✓
All x-values are unique: 2, 3, 4
(Same y-value is okay!)
Answer: YES, this IS a function ✓
Vertical Line Test
Vertical Line Test:
A graph represents a function if and only if NO vertical line intersects the graph at MORE THAN ONE point
How to Use:
1. Imagine drawing vertical lines across the graph
2. If ANY vertical line touches the graph at 2 or more points → NOT a function
3. If EVERY vertical line touches at most 1 point → IS a function
A graph represents a function if and only if NO vertical line intersects the graph at MORE THAN ONE point
How to Use:
1. Imagine drawing vertical lines across the graph
2. If ANY vertical line touches the graph at 2 or more points → NOT a function
3. If EVERY vertical line touches at most 1 point → IS a function
Example 1: Straight Line $y = 2x + 1$
Any vertical line crosses this line at exactly ONE point
Result: IS a function ✓
Any vertical line crosses this line at exactly ONE point
Result: IS a function ✓
Example 2: Circle $x^2 + y^2 = 25$
A vertical line through the middle crosses at TWO points
Result: NOT a function ✗
A vertical line through the middle crosses at TWO points
Result: NOT a function ✗
Example 3: Parabola $y = x^2$
Any vertical line crosses at exactly ONE point
Result: IS a function ✓
Any vertical line crosses at exactly ONE point
Result: IS a function ✓
Quick Check:
• Vertical line (like $x = 3$) → NOT a function
• Horizontal line (like $y = 5$) → IS a function
• Most basic graphs you study are functions!
• Vertical line (like $x = 3$) → NOT a function
• Horizontal line (like $y = 5$) → IS a function
• Most basic graphs you study are functions!
6-8. Find Values & Evaluate Functions
Function Notation: $f(x)$
• $f$ is the name of the function
• $x$ is the input variable
• $f(x)$ is the output (same as y)
Reading: $f(3) = 7$ reads as "f of 3 equals 7"
Meaning: When input is 3, output is 7
• $f$ is the name of the function
• $x$ is the input variable
• $f(x)$ is the output (same as y)
Reading: $f(3) = 7$ reads as "f of 3 equals 7"
Meaning: When input is 3, output is 7
6. Find Values Using Function Graphs
To find $f(a)$ from a graph:
Step 1: Find $x = a$ on the x-axis
Step 2: Draw (or imagine) a vertical line at $x = a$
Step 3: Find where this line intersects the graph
Step 4: Read the y-coordinate of that point
Step 5: That y-value is $f(a)$
Step 1: Find $x = a$ on the x-axis
Step 2: Draw (or imagine) a vertical line at $x = a$
Step 3: Find where this line intersects the graph
Step 4: Read the y-coordinate of that point
Step 5: That y-value is $f(a)$
Example: From a graph, point $(2, 5)$ is on the function
This means: $f(2) = 5$
When $x = 2$, $y = 5$
This means: $f(2) = 5$
When $x = 2$, $y = 5$
7. Evaluate a Function
To Evaluate $f(x)$ at $x = a$:
Step 1: Take the function formula
Step 2: Replace every $x$ with the given value
Step 3: Simplify using order of operations
Step 4: The result is $f(a)$
Step 1: Take the function formula
Step 2: Replace every $x$ with the given value
Step 3: Simplify using order of operations
Step 4: The result is $f(a)$
Example 1: Given $f(x) = 3x + 5$, find $f(2)$
Substitute $x = 2$:
$f(2) = 3(2) + 5$
$f(2) = 6 + 5$
$f(2) = 11$
Answer: $f(2) = 11$
Substitute $x = 2$:
$f(2) = 3(2) + 5$
$f(2) = 6 + 5$
$f(2) = 11$
Answer: $f(2) = 11$
Example 2: Given $f(x) = x^2 - 4x + 1$, find $f(5)$
$f(5) = (5)^2 - 4(5) + 1$
$f(5) = 25 - 20 + 1$
$f(5) = 6$
Answer: $f(5) = 6$
$f(5) = (5)^2 - 4(5) + 1$
$f(5) = 25 - 20 + 1$
$f(5) = 6$
Answer: $f(5) = 6$
Example 3: Given $f(x) = 2x - 3$, find $f(-1)$
$f(-1) = 2(-1) - 3$
$f(-1) = -2 - 3$
$f(-1) = -5$
Answer: $f(-1) = -5$
$f(-1) = 2(-1) - 3$
$f(-1) = -2 - 3$
$f(-1) = -5$
Answer: $f(-1) = -5$
8. Evaluate a Function: Plug in an Expression
When input is an expression (not just a number):
Replace every $x$ with the ENTIRE expression in parentheses
Replace every $x$ with the ENTIRE expression in parentheses
Example 1: Given $f(x) = x + 4$, find $f(2a)$
Replace $x$ with $2a$:
$f(2a) = (2a) + 4 = 2a + 4$
Answer: $f(2a) = 2a + 4$
Replace $x$ with $2a$:
$f(2a) = (2a) + 4 = 2a + 4$
Answer: $f(2a) = 2a + 4$
Example 2: Given $f(x) = x^2 + 2x$, find $f(x + 1)$
Replace every $x$ with $(x + 1)$:
$f(x + 1) = (x + 1)^2 + 2(x + 1)$
$f(x + 1) = x^2 + 2x + 1 + 2x + 2$
$f(x + 1) = x^2 + 4x + 3$
Answer: $f(x + 1) = x^2 + 4x + 3$
Replace every $x$ with $(x + 1)$:
$f(x + 1) = (x + 1)^2 + 2(x + 1)$
$f(x + 1) = x^2 + 2x + 1 + 2x + 2$
$f(x + 1) = x^2 + 4x + 3$
Answer: $f(x + 1) = x^2 + 4x + 3$
Example 3: Given $f(x) = 3x - 1$, find $f(a + 2)$
$f(a + 2) = 3(a + 2) - 1$
$f(a + 2) = 3a + 6 - 1$
$f(a + 2) = 3a + 5$
Answer: $f(a + 2) = 3a + 5$
$f(a + 2) = 3(a + 2) - 1$
$f(a + 2) = 3a + 6 - 1$
$f(a + 2) = 3a + 5$
Answer: $f(a + 2) = 3a + 5$
9-10. Complete Function Tables
9. Complete a Function Table from a Graph
Steps:
Step 1: Look at the x-values in the table
Step 2: For each x-value, find the corresponding y-value on the graph
Step 3: Fill in the y-values in the table
Step 1: Look at the x-values in the table
Step 2: For each x-value, find the corresponding y-value on the graph
Step 3: Fill in the y-values in the table
10. Complete a Function Table from an Equation
Steps:
Step 1: Take each x-value from the table
Step 2: Substitute it into the equation
Step 3: Calculate the y-value
Step 4: Write the y-value in the table
Step 1: Take each x-value from the table
Step 2: Substitute it into the equation
Step 3: Calculate the y-value
Step 4: Write the y-value in the table
Example: Complete the table for $y = 2x - 1$
Calculate:
$x = 0$: $y = 2(0) - 1 = -1$
$x = 1$: $y = 2(1) - 1 = 1$
$x = 2$: $y = 2(2) - 1 = 3$
$x = 3$: $y = 2(3) - 1 = 5$
Completed Table:
| x | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| y | ? | ? | ? | ? |
Calculate:
$x = 0$: $y = 2(0) - 1 = -1$
$x = 1$: $y = 2(1) - 1 = 1$
$x = 2$: $y = 2(2) - 1 = 3$
$x = 3$: $y = 2(3) - 1 = 5$
Completed Table:
| x | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| y | -1 | 1 | 3 | 5 |
11-12. Find & Approximate Solutions Using Tables
11. Find Solutions Using a Table
Solution: An x-value where $f(x)$ equals a specific value
Finding in table: Look for the row where y equals the target value
Finding in table: Look for the row where y equals the target value
Example: Find $x$ when $f(x) = 7$
Look for $f(x) = 7$ → occurs when $x = 3$
Answer: $x = 3$
| x | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| f(x) | 1 | 3 | 5 | 7 | 9 |
Look for $f(x) = 7$ → occurs when $x = 3$
Answer: $x = 3$
12. Approximate Solutions Using a Table
When exact value isn't in table:
• Find two values that surround the target
• The solution is BETWEEN those x-values
• Use interpolation if needed
• Find two values that surround the target
• The solution is BETWEEN those x-values
• Use interpolation if needed
Example: Approximate $x$ when $f(x) = 6$
$f(x) = 6$ is between $f(2) = 5$ and $f(3) = 8$
Answer: $x$ is between 2 and 3 (approximately 2.3 - 2.5)
| x | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| f(x) | 2 | 5 | 8 | 11 |
$f(x) = 6$ is between $f(2) = 5$ and $f(3) = 8$
Answer: $x$ is between 2 and 3 (approximately 2.3 - 2.5)
13-14. Interpret Functions & Identify Graphs
13. Interpret Functions Using Everyday Language
Real-World Interpretation: Explaining what function values mean in context
Example: $h(t)$ represents height of ball (feet) after $t$ seconds
If $h(2) = 64$:
Mathematical: When $t = 2$, $h = 64$
Everyday Language: "After 2 seconds, the ball is 64 feet high"
If $h(2) = 64$:
Mathematical: When $t = 2$, $h = 64$
Everyday Language: "After 2 seconds, the ball is 64 feet high"
Example: $C(n)$ is cost in dollars for $n$ tickets
If $C(5) = 75$:
Meaning: "5 tickets cost $75"
If $C(5) = 75$:
Meaning: "5 tickets cost $75"
14. Identify Graphs: Word Problems
Match situations to graph shapes:
• Increasing: Line/curve going up
• Decreasing: Line/curve going down
• Constant: Horizontal line
• Fast change: Steep slope
• Slow change: Gentle slope
• Increasing: Line/curve going up
• Decreasing: Line/curve going down
• Constant: Horizontal line
• Fast change: Steep slope
• Slow change: Gentle slope
Example: "Temperature rises quickly, then levels off"
Graph shape: Steep increase, then horizontal line
Graph shape: Steep increase, then horizontal line
15-16. Rate of Change
Rate of Change:
How much the output changes per unit change in input
Formula:
$$\text{Rate of Change} = \frac{\text{Change in y}}{\text{Change in x}} = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}$$
Note: For linear functions, rate of change = slope
How much the output changes per unit change in input
Formula:
$$\text{Rate of Change} = \frac{\text{Change in y}}{\text{Change in x}} = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}$$
Note: For linear functions, rate of change = slope
15. Rate of Change: Tables
Steps to Find Rate of Change from Table:
Step 1: Choose two points from the table
Step 2: Find the change in y: $y_2 - y_1$
Step 3: Find the change in x: $x_2 - x_1$
Step 4: Divide: $\frac{y_2 - y_1}{x_2 - x_1}$
Step 5: Simplify
Step 1: Choose two points from the table
Step 2: Find the change in y: $y_2 - y_1$
Step 3: Find the change in x: $x_2 - x_1$
Step 4: Divide: $\frac{y_2 - y_1}{x_2 - x_1}$
Step 5: Simplify
Example 1: Find rate of change
Use points $(1, 5)$ and $(3, 11)$:
$\text{Rate} = \frac{11 - 5}{3 - 1} = \frac{6}{2} = 3$
Answer: 3 units per unit (or slope = 3)
Meaning: For every 1 unit increase in x, y increases by 3
| x | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| y | 5 | 8 | 11 | 14 |
Use points $(1, 5)$ and $(3, 11)$:
$\text{Rate} = \frac{11 - 5}{3 - 1} = \frac{6}{2} = 3$
Answer: 3 units per unit (or slope = 3)
Meaning: For every 1 unit increase in x, y increases by 3
Example 2: Real-world context
$\text{Rate} = \frac{200 - 100}{4 - 2} = \frac{100}{2} = 50$ miles/hour
Meaning: The car travels 50 miles per hour
| Hours | 0 | 2 | 4 | 6 |
|---|---|---|---|---|
| Miles | 0 | 100 | 200 | 300 |
$\text{Rate} = \frac{200 - 100}{4 - 2} = \frac{100}{2} = 50$ miles/hour
Meaning: The car travels 50 miles per hour
16. Rate of Change: Graphs
To Find Rate of Change from Graph:
Step 1: Choose two clear points on the line/curve
Step 2: Find the rise (vertical change)
Step 3: Find the run (horizontal change)
Step 4: Calculate: $\frac{\text{rise}}{\text{run}}$
Step 1: Choose two clear points on the line/curve
Step 2: Find the rise (vertical change)
Step 3: Find the run (horizontal change)
Step 4: Calculate: $\frac{\text{rise}}{\text{run}}$
Example: Points $(1, 3)$ and $(5, 11)$ on a graph
Rise: $11 - 3 = 8$
Run: $5 - 1 = 4$
Rate: $\frac{8}{4} = 2$
Answer: Rate of change = 2
Rise: $11 - 3 = 8$
Run: $5 - 1 = 4$
Rate: $\frac{8}{4} = 2$
Answer: Rate of change = 2
Types of Rate of Change:
• Positive rate: Graph increases (goes up)
• Negative rate: Graph decreases (goes down)
• Zero rate: Graph is horizontal (no change)
• Constant rate: Straight line (same rate throughout)
• Variable rate: Curved line (rate changes)
• Positive rate: Graph increases (goes up)
• Negative rate: Graph decreases (goes down)
• Zero rate: Graph is horizontal (no change)
• Constant rate: Straight line (same rate throughout)
• Variable rate: Curved line (rate changes)
Average Rate of Change:
For non-linear functions, rate of change varies
• Average rate of change = slope of secant line
• Calculated same way: $\frac{y_2 - y_1}{x_2 - x_1}$
• Gives average change over an interval
For non-linear functions, rate of change varies
• Average rate of change = slope of secant line
• Calculated same way: $\frac{y_2 - y_1}{x_2 - x_1}$
• Gives average change over an interval
Quick Reference Guide
Function Definition:
Each input has EXACTLY ONE output
• Test: No repeated x-values with different y-values
• Vertical Line Test: No vertical line crosses graph more than once
Each input has EXACTLY ONE output
• Test: No repeated x-values with different y-values
• Vertical Line Test: No vertical line crosses graph more than once
Key Vocabulary:
• Domain: All x-values (inputs)
• Range: All y-values (outputs)
• Independent Variable: Input (x)
• Dependent Variable: Output (y)
• $f(x)$: Function notation ("f of x")
• Domain: All x-values (inputs)
• Range: All y-values (outputs)
• Independent Variable: Input (x)
• Dependent Variable: Output (y)
• $f(x)$: Function notation ("f of x")
Evaluating Functions:
Given $f(x)$, to find $f(a)$:
1. Replace every $x$ with $a$
2. Simplify
3. Result is $f(a)$
Given $f(x)$, to find $f(a)$:
1. Replace every $x$ with $a$
2. Simplify
3. Result is $f(a)$
Rate of Change Formula:
$$\text{Rate} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\Delta y}{\Delta x}$$
For linear functions: Rate of change = slope
$$\text{Rate} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\Delta y}{\Delta x}$$
For linear functions: Rate of change = slope
| Concept | Key Question | How to Check |
|---|---|---|
| Is it a function? | Does each x have only one y? | Check for repeated x-values or use vertical line test |
| Find domain | What are all possible inputs? | List all x-values |
| Find range | What are all possible outputs? | List all y-values |
| Evaluate function | What is f(a)? | Substitute a for x and simplify |
| Rate of change | How fast is y changing? | Calculate $\frac{\Delta y}{\Delta x}$ |
Four Representations Summary
| Representation | Example | Best For |
|---|---|---|
| Ordered Pairs | $\{(1, 2), (2, 4), (3, 6)\}$ | Listing specific points |
| Table | x: 1, 2, 3 | y: 2, 4, 6 | Organizing data, finding patterns |
| Graph | Points plotted on plane | Visual understanding, trends |
| Mapping | 1→2, 2→4, 3→6 | Showing input-output connections |
Success Tips for Functions:
✓ Remember: Each input → ONE output (function rule)
✓ Domain = inputs (x), Range = outputs (y)
✓ Use vertical line test on graphs
✓ Independent variable = what you control
✓ Dependent variable = what responds/changes
✓ $f(3) = 7$ means "when x=3, y=7"
✓ Rate of change = slope for linear functions
✓ Practice all four representations
✓ Always check if x-values repeat with different y-values
✓ Remember: Each input → ONE output (function rule)
✓ Domain = inputs (x), Range = outputs (y)
✓ Use vertical line test on graphs
✓ Independent variable = what you control
✓ Dependent variable = what responds/changes
✓ $f(3) = 7$ means "when x=3, y=7"
✓ Rate of change = slope for linear functions
✓ Practice all four representations
✓ Always check if x-values repeat with different y-values