Introduction to Functions

What is a Function?

A function is a rule that assigns each input exactly one output.

Think of it as a machine:

• Put in a number (input)
• Machine applies a rule
• Get out a number (output)

Key rule: Each input has EXACTLY ONE output!

Input-Output Tables

Example: "Double the number then add 3"

Pattern: y = 2x + 3

Function Notation

Standard notation: y = 2x + 3

Function notation: f(x) = 2x + 3

Read as: "f of x equals 2x plus 3"

f(x) means "the output when input is x"

Same as y! Just a different way to write it.

Evaluating Functions

To find f(3):

1. Replace x with 3
2. Calculate

Example 1: Evaluate f(5)

Function: f(x) = 2x + 3

Find f(5):

f(5) = 2(5) + 3
f(5) = 10 + 3
f(5) = 13

Answer: f(5) = 13

Meaning: When input is 5, output is 13

Example 2: Evaluate f(0)

Function: f(x) = x² − 4

Find f(0):

f(0) = 0² − 4
f(0) = 0 − 4
f(0) = −4

Answer: f(0) = −4

Example 3: Evaluate f(−2)

Function: f(x) = 3x + 1

Find f(−2):

f(−2) = 3(−2) + 1
f(−2) = −6 + 1
f(−2) = −5

Answer: f(−2) = −5

Identifying Functions

Is it a function? Check if each input has exactly one output.

Example 1: Is This a Function?

Yes, it's a function! Each x-value has exactly one y-value.

Example 2: NOT a Function

Not a function! Input x=2 has TWO outputs (4 and 5)

Domain and Range

Domain: All possible input values (x-values)

Range: All possible output values (y-values)

Example: Find Domain and Range

Function: {(1,3), (2,5), (3,7), (4,9)}

Domain: {1, 2, 3, 4} (all x-values) Range: {3, 5, 7, 9} (all y-values)

Writing Function Rules

Example 1: From Table to Rule

Pattern: y is 4 times x

Rule: f(x) = 4x

Example 2: From Words to Rule

Words: "Square the number then subtract 5"

Rule: f(x) = x² − 5

Example 3: From Pattern

Pattern: Each output is 3 more than twice the input

Rule: f(x) = 2x + 3

Graphing Functions

Each ordered pair (x, y) is a point!

For f(x) = x + 2:

• f(0) = 2 → point (0, 2)
• f(1) = 3 → point (1, 3)
• f(2) = 4 → point (2, 4)

Plot these points and connect!

Vertical Line Test

On a graph: If any vertical line crosses the graph more than once, it's NOT a function.

Why? That would mean one input has multiple outputs!

Example: Using Vertical Line Test

Circle: NOT a function (vertical line crosses twice) Straight line (not vertical): IS a function

Different Types of Functions

Linear: f(x) = mx + b

• Makes a straight line
• Example: f(x) = 2x + 1

Quadratic: f(x) = x²

• Makes a parabola (U-shape)
• Example: f(x) = x² − 3

Constant: f(x) = c

• Always same output
• Example: f(x) = 5

Function vs. Relation

Relation: Any set of ordered pairs

• Can be a function or not

Function: Special relation where each input has ONE output

• All functions are relations
• Not all relations are functions

Real-World Functions

Vending machine: Insert money (input) → get snack (output)

• Each button gives one specific snack

Temperature conversion: F = (9/5)C + 32

• Each Celsius temp gives one Fahrenheit temp

Taxi fare: Cost based on distance

• f(d) = 3 + 2d (d = distance in miles)

Cell phone plan: Cost based on data used

• f(g) = 40 + 10g (g = GB over limit)

Practice Problems

Example: Complete the Table

Function: f(x) = 3x − 2

Solutions:

• f(0) = 3(0) − 2 = −2
• f(2) = 3(2) − 2 = 4
• f(5) = 3(5) − 2 = 13

Practice