Functions

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1. Introduction to Functions

What is a Function?

Function:

A mathematical machine that takes inputs and transforms them into outputs using one or more operations. Every input has exactly one output. Think of it as a rule that describes what happens to a number.

Example: "Double the number and add 1" is a function with two operations:

Input 3 → 2(3) + 1 = 7 (output)
Input −4 → 2(−4) + 1 = −7 (output)
Input x → 2x + 1 (output)

Function Notation: f(x)

f(x) = ... notation:

This is shorthand for describing a function. f is the name of the function, x is the input, and f(x) is the output. You can use any letter (g, h, p, etc.). The part after the equals sign is the rule.

Example: f(x) = 2x + 1 means "the function f takes an input x and outputs 2x + 1"

💡 Key Difference: f(x) = 15 means "the output is 15" (so we need to find what input gives this). But f(15) means "substitute 15 as the input" (find the output when x = 15).

How Functions Work

You have an input: x = 3 (or any value)

You substitute it into the function: f(3) = 2(3) + 1

You calculate the output: f(3) = 7

Example with f(x) = 3x² − 2x + 1:

Find f(7):
f(7) = 3(7)² − 2(7) + 1
= 3(49) − 14 + 1
= 147 − 14 + 1
= 134

You can also substitute an expression (like x + 3) as your input, not just a number.

Mapping Diagrams

A mapping diagram shows how a function transforms specific inputs into outputs. Arrows connect each input to its output. This visualizes the "function as a machine" idea.

2. Domain & Range

What is the Domain?

Domain:

The set of all allowed inputs that you can put into a function. Domains describe x values, never y or f(x) values. Some functions naturally exclude certain inputs (like 1/x excludes x = 0 because you can't divide by zero).

Domain Examples:

f(x) = 1/x: Domain is "all values of x except 0" (x ≠ 0)
f(x) = √x: Domain is "x ≥ 0" (can't take square root of negatives)
f(x) = x²:

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Also in the full note

3. Composite Functions
4. Inverse Functions
5. Common Mistakes & How to Avoid Them
Summary: The Big Picture
Concepts Checklist
What to Memorize
Exam Tips & Tricks
What is the Range?

What's inside

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Functions

Functions

1. Introduction to Functions

What is a Function?

Function Notation: f(x)

How Functions Work

Mapping Diagrams

2. Domain & Range

What is the Domain?

More Mathematics topics