1st PUC Mathematics · Chapter 2

Relations & Functions — Domain, Range and What Actually Makes Something a Function

Relations and Functions is the chapter where 1st PUC mathematics stops being about individual numbers and starts being about how one set connects to another — the idea that quietly underlies calculus, coordinate geometry, and most of what comes after it. Most students memorise the definitions without ever being shown why a function has that one restriction relations don’t. This guide fixes that.

Ordered Pairs and the Cartesian Product

An ordered pair (a, b) is different from (b, a) — order matters, which is the whole point of the word “ordered.” If A and B are two sets, the Cartesian product A × B is the set of all possible ordered pairs (a, b) where a ∈ A and b ∈ B. If A has m elements and B has n elements, A × B has exactly m × n elements — a fact worth internalising rather than memorising, since it follows directly from counting: for every one of the m choices of a, there are n choices of b.

Worked example: If A = {1, 2} and B = {3, 4, 5}, then A × B has 2 × 3 = 6 elements: (1,3), (1,4), (1,5), (2,3), (2,4), (2,5). Note that B × A would give (3,1), (3,2), (4,1), (4,2), (5,1), (5,2) — a completely different set of pairs, because (3,1) ≠ (1,3).

What a Relation Actually Is

A relation R from set A to set B is simply any subset of A × B. That’s the whole definition — no restrictions, no special rules. If A × B has 6 possible pairs, there are 2⁶ = 64 possible relations from A to B, including the empty relation and the full relation A × B itself.

The domain of a relation is the set of all first elements of the ordered pairs actually used. The range is the set of all second elements actually used — not to be confused with the codomain, which is the full set B the relation maps into, whether or not every element of B gets used.

What Makes a Relation a Function

A function is a relation with exactly one extra rule: every element of the domain must map to exactly one element of the range — no element of A is left unmapped, and no element of A maps to more than one element of B. This is the single fact that every exam question on this topic is really testing, even when it’s disguised as a graph, an arrow diagram, or a set of ordered pairs.

The Vertical Line Test: On a graph, if any vertical line drawn through the curve touches it at more than one point, the curve is not a function of x — because that x-value is mapping to more than one y-value. A circle x² + y² = r² fails this test (a vertical line through the middle crosses it twice); a straight line y = mx + c always passes.

Worked example: R = {(1,2), (2,3), (3,2), (4,5)} — is this a function? Every first element (1, 2, 3, 4) appears exactly once, so yes, this is a function, even though the outputs 2 and 5 repeat in the range. Repetition in the range is fine; repetition in the domain mapping to different outputs is what breaks a function.

Now compare: R = {(1,2), (1,3), (2,4)} — is this a function? No. The element 1 appears twice in the domain, mapped to two different outputs (2 and 3). This is a relation, but not a function.

Finding Domain and Range of Real Functions

For real-valued functions given by a formula, the domain is every real x for which the formula gives a defined, real output. Two restrictions come up constantly in exams:

Denominators can’t be zero. For f(x) = 1/(x−3), the domain is all real numbers except x = 3, written as R − {3}.

Square roots need non-negative content. For f(x) = √(x−2), the domain requires x − 2 ≥ 0, so x ≥ 2, written as [2, ∞).

Worked example: Find the domain of f(x) = 1/√(x−4). Two conditions apply at once: the expression under the root must be positive (not just non-negative, since it’s also in a denominator), so x − 4 > 0, giving domain (4, ∞).

Where Students Actually Lose Marks

The most common error is confusing range with codomain — writing down the full target set B as the “range” without checking whether every element of B is actually used by the relation. The second most common error is applying the domain restrictions in the wrong order on nested expressions, like forgetting that a square root inside a denominator needs a strict inequality (> 0), not the ≥ 0 used for a standalone square root.

Frequently Asked Questions

Is every function a relation?

Yes — every function is a relation that happens to satisfy the extra one-output-per-input rule. Every relation is not a function, but every function is a relation.

What’s the difference between range and codomain?

Codomain is the full set the function is allowed to map into; range is the subset of the codomain actually produced by the function. Range is always a subset of the codomain — sometimes equal to it, sometimes smaller.

How does this chapter connect to what comes later?

Domain-finding here is exact preparation for Limits and Derivatives, where every derivative question starts by confirming the function is even defined at the point in question — and for 2nd PUC Continuity and Differentiability, which is built entirely on this foundation.

Build This Foundation Properly

Relations and Functions sets up half the syllabus that follows. Daniel Sir teaches it as a foundation, not a chapter to rush through. See the PUC Mathematics Programme →