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How do the domain and range describe 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 between an independent variable (typically $x$ ) and a dependent variable (typically $y$ ), where each input is associated with exactly one unique output. The domain and range are fundamental sets that describe this relationship:1

Domain: This is the set of all possible values for the independent variable ( $x$ ). It represents the complete collection of inputs for which the function is defined.1
Range: This is the set of all resulting values for the dependent variable ( $f (x)$ ). It consists of all the outputs generated by applying the function's rule to the values within the domain.1

Together, these sets define the scope and behavior of the function. For instance, in analytic geometry, the domain corresponds to the values plotted along the horizontal $x$ -axis, while the range corresponds to the values plotted along the vertical $y$ -axis.1