Definition

1. Noun:
   • A single-valued function is a mathematical relation where each input from a set (called the domain) is associated with exactly one output in another set (called the range). This is the standard definition of a "function" in mathematics.

Usage

• The term single-valued function is used to emphasize the defining property that one input yields only one output, distinguishing it from more general relations (like a circle equation, which can give two outputs for one input).
• It is often used in foundational discussions or in contexts where clarifying the core property of a function is necessary.

Examples

• Noun:
  • In calculus, we primarily study single-valued functions like f(x) = x².
  • The equation y = ±√x does not represent a single-valued function because a positive x-value produces two possible y-values.

Advanced Usage

• The phrase "single-valued" is the critical modifier. A function is, by its modern definition, inherently single-valued. Therefore, "single-valued function" is sometimes considered a redundant phrase, though it is used for absolute clarity.
• Its opposite is a "multi-valued function" (like the complex logarithm), which is technically a relation and not a function in the strictest sense.

Variants and Related Words

• Function (n): The more common and concise term for a single-valued function.
• Map or Mapping (n): Synonyms often used in more abstract or general mathematical contexts.
• Transformation (n): Often used when the function implies a specific change or operation, especially in geometry or linear algebra.

Synonyms

• Function
• Map
• Mapping

Related Terms (Conceptual)

• Domain: The set of all possible inputs for the function.
• Range or Codomain: The set of all possible outputs (range) or the set into which all outputs fall (codomain).
• Relation: A more general pairing of elements from two sets, which may or may not be a function.

Noun

1. (mathematics) a mathematical relation such that each element of a given set (the domain of the function) is associated with an element of another set (the range of the function)