Functions

Chapter Summary

1. Language of Functions

What is a Mapping?

A mapping takes an input from one set of values and produces an output in another set (or the same set). Think of it like a machine: you feed in a number, it does something to it, and spits out a result.

There are four types of mappings:

• One-to-one: Each input maps to exactly one output, and each output comes from exactly one input. (Like a perfect pairing dance.)
• Many-to-one: Multiple inputs can map to the same output. (e.g., squaring: both 2 and −2 give 4.)
• One-to-many: One input maps to multiple outputs. (e.g., y² = x — one x can give two y values.)
• Many-to-many: Multiple inputs map to multiple outputs in a tangled way.

What Makes a Mapping a Function?

This means:

• One-to-one mappings are functions. ✓
• Many-to-one mappings are functions. ✓ (Yes! Many inputs can give one output.)
• One-to-many mappings are NOT functions. ✗
• Many-to-many mappings are NOT functions. ✗

Why? Because in a one-to-many mapping, a single input x produces multiple outputs. That violates the golden rule: "one input, one output."

Function Notation

Functions are written using notation like f(x), g(x), etc. You can write it two ways:

Sets of Numbers: ℕ, ℤ, ℚ, ℝ

Functions work with different types of numbers. Here's the hierarchy:

ℕ (Natural numbers)      1, 2, 3, 4, ...
    ↓
ℤ (Integers)             ..., -2, -1, 0, 1, 2, ...
    ↓
ℚ (Rationals/Quotients)  1/2, 3/4, -5/3, ...
    ↓
ℝ (Real numbers)         All of the above + π, √2, e, ...

ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ — each set contains all the ones before it.

You might also see ℤ⁻ (negative integers only) or ℝ⁺ (positive reals only).

Domain and Range

Crucial: A function is only fully defined when you state its domain. Without a domain, it's incomplete.