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How is a function different from a relation?

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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 relation is broadly defined as a set of ordered tuples (such as pairs, triples, or quadruples) that establishes a connection between objects.1

A function is a specific, more restrictive type of relation. While a relation can associate one input with multiple outputs, a function requires that for every value of an independent variable ( $x$ ), there is exactly one unique value for the dependent variable ( $y$ ).2

In terms of analytic geometry, this distinction is often visualized using the vertical line test: if you plot a relation on an $x$ - $y$ plane, it qualifies as a function only if a vertical line passes through no more than one point on the curve at any given $x$ value. Essentially, while all functions are relations, not all relations are functions because they may lack the requirement of producing a unique output for every input.2