How are multivariable functions used to relate several inputs to one output?

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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 a rule that defines a relationship between variables. While many functions relate a single independent variable to a dependent variable, multivariable (or multivariate) functions relate several independent variables to a single dependent output.1

In these cases, the output is determined by a rule that incorporates multiple inputs simultaneously. A common example is the formula for the area of a triangle, $A = \frac{b h}{2}$ . Here, the dependent variable $A$ (the area) is a function of two independent variables: $b$ (base) and $h$ (height). By assigning specific numerical values to both $b$ and $h$ , the rule produces a unique value for $A$ .1

This concept extends to more complex applications, such as in complex functions, where a function of a complex variable $z = x + i y$ can be expressed by splitting it into real and imaginary parts, each of which acts as a multivariable function of the real variables $x$ and $y$ .1