Question

what defines a function and how can you tell if a graph is one

Answer 1

A function is a rule that assigns to each input exactly one output. More formally, a function f from a set A (the domain) to a set B (the codomain) is a set of ordered pairs (x, y) with x in A and y in B such that each x in A appears in exactly one pair.

How to tell from a graph whether it represents y as a function of x:
- Use the vertical line test: if any vertical line x = c intersects the graph in more than one point, the graph does NOT represent y as a function of x. If every vertical line intersects the graph in at most one point, it does represent a function.
- Why it works: a vertical line fixes an x-value. Multiple intersection points mean that x has more than one y-value, which violates the “exactly one output” rule.

Quick examples:
- y = x^2 (parabola opening up) is a function: each x has one y.
- Circle x^2 + y^2 = 1 is not a function of x (vertical lines generally meet it twice).
- A vertical line x = 2 is not the graph of a function y = f(x) (the single x would correspond to many y’s).
- The horizontal line test is different: it checks whether a function is one-to-one (injective). If any horizontal line meets the graph more than once, the function is not one-to-one.

Note: Graphs of relations that are not functions can sometimes be turned into functions by restricting the domain or choosing one value (e.g., the top semicircle y = +sqrt(1 − x^2) is a function).