1. What is a Function?

Definition: A function is a special relationship where each input (x-value) has exactly one output (y-value).

Function Notation:

\( f(x) \) or \( y = f(x) \)

Read as "f of x" where x is the input and f(x) is the output

Key Characteristics:

• One-to-one or many-to-one: Each input has exactly ONE output
• NOT one-to-many: One input cannot have multiple outputs
• Unique mapping: If x = a, then f(a) must be a single value

Example:

Function: {(1, 2), (2, 4), (3, 6), (4, 8)}

✓ Each input has exactly one output

NOT a Function: {(1, 2), (1, 3), (2, 4), (3, 5)}

✗ Input 1 has two outputs (2 and 3)

2. Identify Functions

From a Set of Ordered Pairs:

Check if any x-value repeats with different y-values.

• If NO x-value repeats → It IS a function
• If an x-value repeats with different y-values → It is NOT a function

From a Table:

1. Look at the input column (usually x)
2. Check if any input value appears more than once
3. If the same input has different outputs → NOT a function
4. If each input has only one output → It IS a function

Examples:

Example 1: Is this a function? {(2, 5), (3, 7), (4, 9), (5, 11)}

All x-values are different: 2, 3, 4, 5

✓ Yes, this IS a function

Example 2: Is this a function? {(1, 4), (2, 5), (1, 6), (3, 7)}

x = 1 appears twice with outputs 4 and 6

✗ No, this is NOT a function

Example 3: Table

✓ Each x-value appears only once → This IS a function

3. Identify Functions: Graphs (Vertical Line Test)

Vertical Line Test:

If any vertical line intersects the graph at MORE THAN ONE POINT, it is NOT a function.

How to Apply the Test:

1. Imagine drawing vertical lines (parallel to y-axis) 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 the graph at most ONCE → It IS a function

Examples:

Graphs that ARE functions:

• Linear graphs (straight lines) - except vertical lines
• Parabolas opening up or down (\( y = x^2 \))
• Cubic functions (\( y = x^3 \))
• Exponential functions (\( y = 2^x \))

Graphs that are NOT functions:

• Circles (\( x^2 + y^2 = r^2 \)) - vertical line crosses twice
• Sideways parabolas (\( x = y^2 \)) - vertical line crosses twice
• Vertical lines (\( x = k \)) - infinite intersections

Why It Works:

A vertical line represents all points with the same x-value. If it crosses the graph multiple times, that means one x-value has multiple y-values, which violates the definition of a function.

4. Identify Independent and Dependent Variables

Definitions:

Independent Variable (Input): The variable that you change or control; usually x

• Also called: input, domain, x-variable
• Plotted on the horizontal axis (x-axis)
• The cause in a cause-effect relationship

Dependent Variable (Output): The variable that depends on the independent variable; usually y

• Also called: output, range, y-variable, f(x)
• Plotted on the vertical axis (y-axis)
• The effect in a cause-effect relationship

How to Identify:

Ask: "Which variable depends on the other?"

The dependent variable changes BECAUSE of the independent variable.

Examples:

Example 1: The number of hours you study affects your test score.

Independent: Hours studied (you control this)

Dependent: Test score (depends on hours studied)

Example 2: The cost of apples depends on how many pounds you buy.

Independent: Pounds of apples (x)

Dependent: Total cost (y)

Example 3: The distance a car travels depends on time.

Independent: Time (t)

Dependent: Distance (d)

Equation: \( d = 60t \) (if traveling at 60 mph)

Example 4: The area of a square depends on its side length.

Independent: Side length (s)

Dependent: Area (A)

Equation: \( A = s^2 \)

5. Find Values Using Function Graphs

Finding f(a) from a Graph:

1. Locate the value 'a' on the x-axis
2. Draw a vertical line from x = a until it hits the graph
3. From that point, draw a horizontal line to the y-axis
4. Read the y-value → This is f(a)

Finding x when f(x) = b:

1. Locate the value 'b' on the y-axis
2. Draw a horizontal line from y = b until it hits the graph
3. From that point, draw a vertical line to the x-axis
4. Read the x-value → This is your answer

Examples:

Example 1: If a graph passes through point (3, 7), find f(3).

When x = 3, y = 7

f(3) = 7

Example 2: If f(5) = 12, what point is on the graph?

When x = 5, f(x) = 12

Point: (5, 12)

Example 3: A graph passes through (0, 2), (1, 5), (2, 8). Find f(0), f(1), and f(2).

f(0) = 2

f(1) = 5

f(2) = 8

6. Complete a Table for a Function Graph

Steps:

1. Look at the graph and identify clear points
2. For each x-value in the table, find the corresponding y-value from the graph
3. Read coordinates carefully (x, y)
4. Fill in the missing values

Tips:

• Look for points where the graph crosses grid lines
• Use a ruler or straight edge if needed
• Check your work by plotting the points

Example:

A line passes through (0, 1), (1, 3), (2, 5), (3, 7). Complete the table:

Pattern: The function is \( y = 2x + 1 \)

7. Domain and Range of Functions

Definitions:

Domain: The set of all possible INPUT values (x-values)

• All x-values that the function can accept
• The set of independent variable values
• Answers the question: "What x-values are allowed?"

Range: The set of all possible OUTPUT values (y-values)

• All y-values that the function can produce
• The set of dependent variable values
• Answers the question: "What y-values result?"

From a Set of Ordered Pairs:

Domain: List all x-coordinates (inputs)

Range: List all y-coordinates (outputs)

From a Graph:

Domain: Look at all x-values from left to right (horizontal extent)

Range: Look at all y-values from bottom to top (vertical extent)

Interval Notation:

• \( [a, b] \) means "from a to b, including a and b"
• \( (a, b) \) means "from a to b, not including a or b"
• \( (-\infty, \infty) \) means "all real numbers"
• \( [a, \infty) \) means "from a to infinity, including a"

Examples:

Example 1: Find domain and range: {(1, 5), (2, 7), (3, 9), (4, 11)}

Domain: {1, 2, 3, 4} (all x-values)

Range: {5, 7, 9, 11} (all y-values)

Example 2: Function \( y = x^2 \)

Domain: All real numbers, \( (-\infty, \infty) \)

Range: \( y \geq 0 \), or \( [0, \infty) \) (parabola opens upward, minimum y = 0)

Example 3: Function \( y = \sqrt{x} \)

Domain: \( x \geq 0 \), or \( [0, \infty) \) (can't take square root of negative)

Range: \( y \geq 0 \), or \( [0, \infty) \) (square root is always non-negative)

Example 4: A graph goes from x = -2 to x = 5, and from y = 1 to y = 8

Domain: [-2, 5]

Range: [1, 8]

8. Common Functions and Their Properties

Quick Reference: Function Concepts

Key Definitions:

• Function: Each input has exactly ONE output
• Domain: Set of all possible inputs (x-values)
• Range: Set of all possible outputs (y-values)
• Independent Variable: Input (x)
• Dependent Variable: Output (y or f(x))

Tests:

• Vertical Line Test: If any vertical line intersects graph more than once → NOT a function
• From ordered pairs: If an x-value repeats with different y-values → NOT a function

Function Notation:

• \( f(x) \) means "the value of f at x" or "f of x"
• If \( f(3) = 7 \), the point (3, 7) is on the graph
• \( y = f(x) \) means y equals the function value at x

💡 Key Tips for Function Concepts

• ✓ Function = each input has ONLY ONE output
• ✓ Vertical Line Test: crosses graph once = function
• ✓ Domain = all possible x-values (inputs)
• ✓ Range = all possible y-values (outputs)
• ✓ Independent variable = x (what you control)
• ✓ Dependent variable = y (what changes as result)
• ✓ f(3) = 7 means when x = 3, y = 7
• ✓ Point (a, b) on graph means f(a) = b
• ✓ Check tables: look for repeated x-values
• ✓ Domain from graph: left to right extent
• ✓ Range from graph: bottom to top extent
• ✓ Circle is NOT a function (fails vertical line test)