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How can a mapping diagram show whether a relation 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.

In mathematics, a function is defined as a relationship where every element in the domain (the independent variable) corresponds to exactly one unique value in the range (the dependent variable).

A mapping diagram visually represents this relationship by using arrows to connect elements from one set to another. To determine if a relation is a function using this diagram, you must observe the arrows originating from the domain:

Function: If every element in the domain has exactly one arrow pointing to an element in the range, the relation is a function. It is acceptable for multiple elements in the domain to point to the same element in the range, as each input still produces a single, unique output.
Not a Function: If any single element in the domain has more than one arrow pointing to different elements in the range, the relation is not a function, as the input does not produce a unique output.

Essentially, the mapping diagram confirms the definition that for every $x$ , there is a unique value of $y$ .1