Definition

1. Noun:
   • A function that reverses another: An inverse function is a mathematical function that undoes the operation of a given function. If a function f maps an input x to an output y, then its inverse function, denoted f⁻¹, maps y back to x.
   • The result of swapping dependent and independent variables: Formally, for two functions f and g to be inverses, the equations f(x) = y and g(y) = x must both hold true for all x in the domain of f and all y in the domain of g.

Usage and Examples

• Noun:
  • The natural logarithm is the inverse function of the exponential function.
  • To find the inverse function, you solve the equation y = f(x) for x.
  • If f(x) = 2x + 3, then its inverse function is f⁻¹(x) = (x - 3)/2.

Advanced Usage

• "to have an inverse function": Describes a function for which an inverse exists. A function must be one-to-one (bijective on its domain) to have a proper inverse function.
  • A quadratic function like f(x) = x² does not have an inverse function over all real numbers unless its domain is restricted.
• "to be the inverse of": Expresses the inverse relationship between two specific functions.
  • The function g(x) = ³√x is the inverse of f(x) = x³.

Variants and Related Words

• Inverse (adjective/noun): Pertaining to or being the inverse. As a noun, it can refer to the inverse element or function in a general sense.
  • The inverse operation of addition is subtraction.
• Invert (verb): To turn upside down or reverse the order, position, or relationship of things. In mathematics, it often means to find the inverse.
  • You can invert the fraction to solve the equation.

Synonyms

• Reverse function: A less formal term emphasizing the reversal of mapping.
• Antifunction: A rare, non-standard synonym.

Important Notes (Not a Phrasal Verb or Idiom)

• The concept of an inverse function is foundational in algebra and calculus. A key property is that and , wherever these compositions are defined.
• The notation specifically denotes the inverse function and does mean (which is the reciprocal).

Noun

1. a function obtained by expressing the dependent variable of one function as the independent variable of another; f and g are inverse functions if f(x)=y and g(y)=x