Definition

1. Noun:
   • (Mathematics) The set of all possible output values produced by a function: The "range of a function" is the complete collection of values that result from applying the function to every element in its domain. It is the set of values of the dependent variable (usually y or f(x)) for which the function is defined when considering its entire input domain.

Usage and Examples

• Noun:
  • To fully understand a function, you must determine both its domain and its range of a function.
  • The range of a function f(x) = sin(x) is the set of real numbers between -1 and 1, inclusive.

Advanced Usage and Notes

• "Range" vs. "Range of a Function": In mathematics, the term "range" can sometimes be ambiguous, as it may refer to the codomain (the set of all possible output values the function theoretically produce). However, the precise phrase "range of a function" (or sometimes "image") specifically denotes the set of output values obtained from the given domain.
• Finding the Range: Determining the range of a function often involves analyzing the function's behavior, such as identifying its maximum and minimum values or understanding its asymptotic behavior.

Variants and Related Terms

• Image (of a function): A synonym for the range of a function. It is the set of all values where is in the domain.
• Codomain: A related but distinct concept. The codomain is the set into which the function's outputs are constrained to fall, while the range of a function is the actual subset of the codomain that is achieved.
• Domain: The set of all possible input values (independent variable) for which the function is defined. This is the counterpart to the range.

Synonyms

• Image
• Range (in its precise mathematical sense, though this can be context-dependent)

Related Phrases and Concepts

• Pre-image (or Inverse Image): The set of all inputs that map to a given output or set of outputs in the range.
• Surjective Function (Onto Function): A function whose range of a function is equal to its codomain. For every element in the codomain, there is at least one input that maps to it.

Noun

1. (mathematics) the set of values of the dependent variable for which a function is defined
   • the image of f(x) = x^2 is the set of all non-negative real numbers if the domain of the function is the set of all real numbers