How can ordered pairs represent a function?

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A graph of the function f(x) = x cubed minus 3x plus 2 is shown on a blue background, with a red curve passing through key points. Local maximum and minimum points are marked with black dots at x equals negative 1 and x equals 1, illustrating important features of the curve. — function Plot of the cubic equation f(x) = x³ − 3x + 2. The plotted points are where changes in curvature occur.

In mathematics, a function defines a relationship where each element from a set (the domain) is associated with a unique value in another set (the range).

Ordered pairs represent this relationship by pairing an independent variable $x$ with its corresponding dependent value $f (x)$ . Specifically, the graph of a function consists of a set of points with coordinates $(x, y)$ , where $y = f (x)$ .

For a collection of ordered pairs to represent a function, the rule of uniqueness must be satisfied: for every $x$ in the domain, there must be exactly one corresponding value of $y$ . If the same $x$ value is associated with more than one $y$ value, the set of ordered pairs does not constitute a function.1