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How does a mathematical function assign an output to each input?

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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 between an independent variable (the input) and a dependent variable (the output) through a specific rule or law.1

The process of assigning an output to an input works as follows:

Rule-based Assignment: For every numerical value assigned to an independent variable $x$ , there is a defined rule that determines a unique value for the dependent variable $y$ .1
Uniqueness: A fundamental requirement of a function is that for every input $x$ , there must be exactly one corresponding output $f (x)$ . An input cannot result in more than one value.1
Mapping: In the language of set theory, a function acts as a mapping, where each element from a set known as the domain is assigned to a particular element in another set, known as the range.21

When represented graphically in the $x$ - $y$ plane, this relationship is verified by the vertical line test: a relation is a function only if any vertical line passes through no more than one point on the curve.1