OneVariable

In mathematics, the term single-variable (also written as one-variable, and occasionally seen as onevariable in informal usage) describes quantities, functions, or equations that depend on exactly one independent variable. This contrasts with multivariable contexts, in which more than one independent variable appears. The most common setting is a real-valued function f of a single real variable x.

Examples include polynomials, exponentials, logarithms, and trigonometric or rational functions in x. A typical single-variable function is f(x) = x^2 + 3x − 2, defined for x in its domain. The domain is the set of x-values for which the function is defined; the range is the corresponding set of y-values.

Graphically, a single-variable function is drawn on the plane with the horizontal axis representing the independent variable and the vertical axis the dependent variable. The behavior of such functions is studied in algebra and calculus and can be described using properties like continuity, monotonicity, extrema, and asymptotics.

In single-variable calculus, one investigates derivatives, integrals, limits, and series with respect to the single variable. The fundamental theorem of calculus links differentiation and integration. Techniques include substitution, the chain rule, and various integration methods.

Single-variable problems appear across disciplines, including physics for motion along a line, biology for growth models, economics for cost or revenue as a function of quantity, and computer science in analyses of one-dimensional data. The concept also underpins more advanced topics in multivariable analysis when additional variables are introduced.