Definition

1. Noun:
   • A mathematical function is a precise rule or relationship that assigns exactly one output value from a set (called the range or codomain) to each input value from another set (called the domain). It describes how one quantity depends on or is determined by another.

Usage

• The core concept is the unique association of one output for every input.
• It is often represented by symbols like , where denotes the function and is an input from the domain.
• Functions can be described using formulas, graphs, tables, or words.

Examples

• Noun:
  • In the equation f(x) = 2x + 1, the mathematical function f doubles the input and then adds one.
  • The relationship between the side length of a square and its area is a mathematical function.
  • A graph showing temperature over time represents a mathematical function where time is the input and temperature is the output.

Advanced Usage

• "To define a function": To formally specify the rule, domain, and range of a mathematical function.
  • We can define a function that maps each student to their test score.
• "To evaluate a function": To compute the output of a function for a specific input.
  • To evaluate the function f(x) at x=3, substitute 3 for x in the formula.
• "A function of": Used to express that one variable depends on another.
  • The distance traveled is a function of both speed and time.

Variants and Related Words

• Function (n): The more general, abbreviated term for "mathematical function" in technical contexts.
  • The concept of a function is fundamental to calculus.
• Mapping (n): A term often used synonymously with "function," emphasizing the idea of associating elements from one set to another.
  • The function describes a mapping from real numbers to real numbers.
• Functional (adj): Pertaining to or having the nature of a function.
  • They studied the functional relationship between variables.

Synonyms

• Mapping: (In mathematics) A correlation between sets where each element of the first set is associated with an element of the second set.
• Transformation: Often used when the function implies a change or operation on the input.

Related Phrases

• Domain of a function: The complete set of all possible input values for the function.
  • The domain of the function √x is all non-negative real numbers.
• Range of a function: The complete set of all possible output values produced by the function.
  • The range of the function x² is all non-negative real numbers.
• Continuous function: A function whose graph has no breaks, jumps, or holes.
• Linear function: A function of the form , whose graph is a straight line.

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)