One Input, One Answer
A relation is a function exactly when each input produces one and only one output; f(x) names that single output. · 11 min
Last folio gave every point an address. Now look at what a table of points can do. A vending code returns one snack. A student ID returns one student. Each pairing is a rule: input goes in, output comes out. Some rules are dependable — ask with the same input twice, and you get the same answer both times. Mathematics reserves a special name for exactly that kind of dependability, and this folio is about earning the right to use it.
Guess before you learn
A table pairs inputs with outputs: (2, 4), (3, 9), (2, 10). Could one dependable rule have produced this table?
Ask the rule about 2 and it says 4; ask again and it says 10. A dependable rule cannot change its mind about the same input — and that single requirement is the entire definition you are about to meet. If you said yes, keep the pencil mark: spotting the repeated input is a skill, not an instinct.
9–12
3–5
A function is a rule that keeps its promise: feed it an input, and it hands back exactly one output. Feed it the same input tomorrow and it hands back the same output. The doubling rule takes 3 and gives 6 — never 6 on Monday and 8 on Tuesday.
A table breaks the promise only one way: the same input showing up twice with two different outputs. Check the input column first — that is where the trouble hides.
6–8
A relation is any collection of input–output pairs. A function is a relation that keeps the rule of one answer: each input appears with exactly one output. To check a table, scan the inputs — if no input repeats with different outputs, it is a function. Outputs may repeat freely: two inputs are allowed to share an answer, but one input is never allowed to give two.
The domain is the collection of inputs the rule accepts; the range is the collection of outputs it actually produces. For the pairs (1, 5), (2, 7), (3, 5): the domain is 1, 2, 3 and the range is 5, 7 — and it is a function, even though 5 appears twice.
9–12
A relation pairs inputs with outputs; a function is a relation in which each input has exactly one output. On a graph, an input is a vertical position: every point directly above or below x = 2 claims 2 as its input. So the vertical line test is the definition made visible — if any vertical line crosses the graph more than once, one input owns two outputs, and the relation is not a function.
The domain is the set of allowed inputs, the range the set of produced outputs. Notice the asymmetry: two inputs may share one output — a horizontal line crossing twice is perfectly legal — but one input may never hold two outputs. Function-ness is a promise about inputs only.
K–2
Every kid in class has one cubby. Say a name, and the rule points to one cubby. Same name tomorrow, same cubby. That is a good rule.
Now a broken rule: Mia gets two cubbies. Say Mia — which one? A good rule gives one answer for each name, never two.
Undergrad
Formally, a function f from X to Y is a set of ordered pairs f ⊆ X × Y such that every x ∈ X appears in exactly one pair. Under this definition the graph is not a picture of the function — the graph is the function. The vertical line test is the uniqueness clause read geometrically: each vertical line x = a meets the graph at most once.
Distinguish codomain from range: the codomain Y is the declared target; the range f(X) is what is actually hit. School tables blur this, but the distinction matters the moment you ask whether a function is onto.
Postgrad
A function is a triple (X, Y, Γ) with Γ ⊆ X × Y total and single-valued. Dropping totality yields partial functions; dropping single-valuedness yields relations — the two axioms fail independently, which is why students need two separate counterexamples. The vertical line test asserts that the fiber of Γ over each x ∈ X is a singleton.
The recurring deep question is well-definedness: whenever an object is defined via a representative — a fraction, a coset, an equivalence class — one must verify the output does not depend on the representative chosen. Every such proof is the Algebra I table check, performed on an infinite table.
function
A relation in which each input produces exactly one output. Two inputs may share an output; one input may never hold two.
A graph is a relation drawn: every plotted point is one input–output pair. To test it, sweep an imaginary vertical line across the picture. Each vertical line marks one input — if the line ever crosses the graph twice, that input holds two outputs at once, and the relation fails. A circle fails immediately. A straight, non-vertical line passes everywhere. The test is not a new rule; it is the definition, checked with a ruler instead of a table.
Now the notation. Write f(x) = 2x + 1 and read it f of x: f names the rule, x names the input, and f(x) names the one output the rule promises. So f(3) is both an instruction — feed 3 to f — and a number: f(3) = 2(3) + 1 = 7. One warning, because most people trip here exactly once: f(x) is not f times x. The parentheses here mark the input, not a product.
Why is this true?
Why is f(3) a single settled number, rather than several possibilities?
Because f is a function: the definition guarantees each input exactly one output. The notation f(3) is only trustworthy because that promise holds.
Evaluate f(x) = 3x − 4 at x = 5 — the steps fade as you master them
f(5) = 3(5) − 4
f(5) = 15 − 4
f(5) = 11
You now hold the working vocabulary of this unit: relation, function, domain, range, and the notation f(x). Nearly every graph in the folios ahead is a function wearing coordinates. Next comes the single number that tells you how fast a line climbs.
Note
If plotting pairs still feels slow, revisit folio 9 before continuing — the vertical line test leans entirely on reading coordinates at speed.
Practice — new ink and old, interleaved
1.Without looking back: what makes a relation a function, and what do domain and range name?
A relation is a function when each input has exactly one output; the domain is the inputs allowed, the range the outputs produced.
How close were you? Grade yourself honestly — it sets your review date.
2.h(x) = x² − 1. Find h(4).
3.Put the table check in working order.
- Compare the outputs attached to any repeated input
- Scan the input column for a value that repeats
- Two different outputs for one input: not a function
4.From memory: what does each number in (x, y) measure, and from where?
The first number measures signed horizontal distance from the origin along the x-axis; the second measures signed vertical distance along the y-axis.
How close were you? Grade yourself honestly — it sets your review date.
5.Which relation is a function?
6.12 ÷ 3 × 2 = ?
7.From memory: what is a literal equation, and how do the balance moves treat the other letters?
An equation made mostly of letters, like a formula; solving it for one letter uses ordinary balance moves while treating every other letter as a number whose value has not been told.
How close were you? Grade yourself honestly — it sets your review date.