Functions · step 7 of 13

Restriction, extension & piecewise definitions

The restriction f|_X keeps the same rule on a smaller domain, and it can gain properties the original lacked — x² is not injective on ℝ but is on [0,∞). An extension goes the other way and must agree with the original everywhere it was already defined. Piecewise definitions are functions built from several rules, and they are legitimate only if the pieces agree wherever they overlap.

Practise restriction, extension & piecewise definitions free → All Functions subtopics

✓ Unlimited questions · marked criterion by criterion · no card needed

Method: how to approach it

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

Restrict by shrinking the domainf|_X(a) = f(a) for a ∈ X. Nothing about the rule changes; the image can only shrink.
Check gained propertiesInjectivity is often gained; surjectivity is often lost. Recheck both against the new domain.
Extend consistentlyAny extension g must satisfy g|_A = f. Where the new domain is disjoint from A the values are free.
Verify piecewise agreementAt every boundary shared by two clauses, both must give the same value, or the definition is not a function.

Worked example

Is f(x) = x² for x ≤ 1 and f(x) = 2x − 1 for x ≥ 1 a well-defined function on ℝ?

The two clauses overlap at exactly x = 1.
First clause at x = 1: 1² = 1.
Second clause at x = 1: 2(1) − 1 = 1.
The values agree, so no point receives two different outputs.

Answer. Yes, it is well defined — and continuous at the join, since the two pieces agree at x = 1.

Where marks get dropped

These are the specific errors that cost credit on restriction, extension & piecewise definitions questions — QED's rubric penalises each of them separately.

Writing overlapping clauses with conflicting values, which defines a relation rather than a function.
Leaving a gap in the domain. Clauses "x < 1" and "x > 1" omit x = 1 entirely, so the function is undefined there.
Assuming agreement at the join implies differentiability. It gives continuity at best; the derivatives may still disagree.

Practise this until it is automatic

Unlimited fresh questions

QED generates new restriction, extension & piecewise definitions problems on demand at warm-up, exam and challenge level, so you can drill this one skill until it stops costing you marks.

Marked like an examiner

Every answer is scored against a point-by-point rubric with partial credit, so you see exactly which step of the method broke down — not just a tick or a cross.

Answer in real notation

A one-tap symbol palette, a visual equation editor and a truth-table builder — or photograph your handwritten working and QED converts it to LaTeX.

Saved to your library

Every question you generate is kept and re-takeable as a timed exam, and your Functions mastery is tracked so you know when this is exam-ready.

Restriction, extension & piecewise definitions — frequently asked questions

Does restriction preserve injectivity?

Yes, always — fewer inputs can only reduce collisions. It does not preserve surjectivity, since the image shrinks.

Is an extension unique?

Almost never. Outside the original domain the values can be anything, so extra conditions like continuity or linearity are needed to pin one down.

How should I write piecewise functions in an exam?

One clause per line with an explicit, exhaustive and non-conflicting condition. Overlaps must be checked, and the union of the conditions must be the whole domain.

The rest of Functions

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

Ready to make restriction, extension & piecewise definitions exam-proof?

Generate your first questions free — no card, no setup, no personal data stored. Practise until the method is second nature.

Start practising free →