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How can a table of values show whether a relationship is 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.

To determine if a relationship is a function using a table of values, you must verify that every input value (the independent variable, typically x) corresponds to exactly one unique output value (the dependent variable, typically y).

In a table, this means that no single value in the input column can be associated with more than one different value in the output column. If you find an input value that appears more than once in the table but is paired with different output values, the relationship fails the definition of a function, as it does not produce a unique result for that input.1

If an input value repeats but is paired with the same output value every time, the relationship may still be considered a function, provided that all other input values also map to a single, unique output.1