What Makes a Function a Function?

Read the explanation, try the on-paper prompts, then explain the idea in your own words. Use AI feedback as a mentor, not a shortcut.

10–20 min focusProof-first mindset

Best flow: read → think on paper → write a short explanation → refine with feedback.

Reading

Core explanation

A function f from a set X to a set Y (written f : X → Y) assigns to each element x ∈ X exactly one element f(x) ∈ Y.

Key pieces:

Domain: the set X of inputs.
Codomain: the set Y of allowed outputs.
Image (range): the set { f(x) : x ∈ X } ⊆ Y.

Important: A function is not just a formula; it is a rule with a specified domain and codomain.

Examples:

f : ℝ → ℝ given by f(x) = x².
g : ℤ → ℤ given by g(n) = 2n + 1.

When proving things about functions, you must be precise about:

Which set the input comes from.
Which set the output lives in.
How the rule behaves for all elements of the domain.

TL;DR — key idea

A function is a rule assigning each element of the domain exactly one element of the codomain; domain and codomain matter in proofs.

Try these in your notebook

Don’t skip this – writing proofs or explanations on paper is where most of the learning actually happens.

1
In your own words, explain why "a function must give exactly one output for each input." Then give an example of a relation that is NOT a function and explain why it fails.

Once you’ve sketched some ideas, summarize the main insight in the reflection box on the right.

Check your understanding

In 3–6 sentences, explain the core idea of this subsection as if you were teaching a friend who hasn’t seen it. Focus on the logic, not just the final statements.

AI is optional. Use it to spot gaps and sharpen your wording, not to replace your own thinking.