Functions · step 10 of 13

Well-definedness of a proposed function

A rule defines a function only if it assigns exactly one output to every input. Two things can go wrong: the rule may give no value somewhere (failing totality), or several values (failing single-valuedness). The subtle case is a rule defined via a representative — on fractions or equivalence classes — where different representatives of the same input might produce different outputs.

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Method: how to approach it

The order below is what examiners expect to see, and each step carries its own marks.

Check totalityEvery element of the declared domain must receive a value. Division and square roots are the usual culprits.
Check single-valuednessNo input may receive two values. Rules involving ± or "choose any" are suspect.
Check representative-independenceIf the input is a class or a fraction, take two representatives of the same input and confirm the outputs agree.
Refute with a concrete clashGive two representatives of one input and two different outputs. That is a complete disproof.

Worked example

Is g : ℚ → ℤ defined by g(p/q) = p + q well defined?

Take the rational number 1/2, represented as 1/2 and also as 2/4.
Using p/q = 1/2: g = 1 + 2 = 3.
Using p/q = 2/4: g = 2 + 4 = 6.
The same input gives two different outputs.

Answer. Not well defined — the value depends on the chosen representative. Restricting to fractions in lowest terms with q > 0 would fix it.

Where marks get dropped

These are the specific errors that cost credit on well-definedness of a proposed function questions — QED's rubric penalises each of them separately.

Assuming any formula defines a function. If the input has multiple representations, the formula must be checked against all of them.
Overlooking totality. f(x) = 1/x is not a function ℝ → ℝ, though it is one on ℝ \ {0}.
Confusing well-definedness with injectivity. They are unrelated: well-definedness is about outputs being unique per input, injectivity about inputs being unique per output.

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Well-definedness of a proposed function — frequently asked questions

Why do quotient constructions always need this check?

Because the elements are classes, and any formula written with a representative might depend on the choice. The check is the price of working with quotients.

Is well-definedness the same as being a function?

Yes — it is the verification that the proposed rule satisfies the definition of a function. The phrase is used when the check is non-obvious.

How do I fix a rule that is not well defined?

Either restrict to canonical representatives (lowest terms, remainders in {0,…,n−1}) or redefine so the output cannot depend on the choice.

The rest of Functions

Injective, surjective, bijective, composition, inverse. Each subtopic below has its own method, worked example and mark-losing traps.

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