Function

Definition

A function maps each allowed input to exactly one output.

Basic form:

input -> output

Example:

f(x) = x + 1

Then:

f(2) = 3
f(5) = 6

Function = rule that assigns one output to each allowed input.

---

Function As A Relation

A function can be seen as a special kind of relation.

Example:

f = {(1, 2), (2, 3), (3, 4)}

Each ordered pair connects an input to an output.

The first value is the input.

The second value is the output.

A function is a relation with a stricter rule.

---

The One Output Requirement

For a relation to be a function, one input cannot point to two different outputs.

This is a function:

{(1, a), (2, b), (3, c)}

This is not a function:

{(1, a), (1, b), (2, c)}

Why?

Because input 1 maps to both a and b.

A function cannot give two different answers for the same input.

---

Inputs Do Not Have To Be Numbers

Functions can operate on many kinds of inputs.

Examples:

lowercase word -> uppercase word
file name -> file extension
person -> birth year
state -> next state

The key requirement is not that the input is numeric.

The key requirement is that each allowed input has exactly one output.

Function is about mapping, not just arithmetic.

---

Why Functions Matter

Functions make transformation precise.

They let us ask:

• what inputs are allowed?
• what output does each input produce?
• can two inputs produce the same output?
• does every possible output get used?

These questions become important when studying structure and computation.

Functions turn rules into precise input-output mappings.

---

Final Summary

• A function maps each allowed input to exactly one output.
• A function can be represented as ordered pairs.
• A function is a special kind of relation.
• One input cannot map to two different outputs.
• Inputs do not have to be numbers.
• Functions make transformation precise.