Functions

What is a Function?

A function is a special type of relation where each element of the domain is associated with exactly one element of the codomain. This means one input gives only one output.

• If f: A → B, then every element of A has a unique image in B
• A = Domain (input values)
• B = Codomain (possible outputs)
• Actual outputs are called Range

Example:

If f(x) = x² and A = {1,2,3}

Then f(A) = {1,4,9}

Key Idea to Remember

• One input → one output (✔ function)
• One input → multiple outputs (❌ not a function)
• Multiple inputs → same output (✔ allowed)

Valid Function (Arrow Diagram)

Each element of domain has exactly one image.

Each input has exactly one output → valid function

Many-to-One Function

Different inputs can have same output. This is still a function.

Multiple inputs → same output (✔ valid function)

Not a Function

If one input has more than one output, it is not a function.

One input → two outputs (❌ not a function)