Concept
Inverse function
Overview
In mathematics, the inverse function of a function f (also called the inverse of f) is a function that undoes the operation of f. The inverse of f exists if and only if f is bijective, and if it exists, is denoted by
f
−
1
.
{\displaystyle f^{-1}.}
For a function
f
:
X
→
Y
{\displaystyle f\colon X\to Y}
, its inverse
f
−
1
:
Y
→
X
{\displaystyle f^{-1}\colon Y\to X}
admits an explicit description: it sends each element
y
∈
Y
{\displaystyle y\in Y}
to the unique element
x
∈
X
{\displaystyle x\in X}
such that f(x) = y.
As an example, consider the real-valued function of a real variable given by f(x) = 5x − 7. One can think of f as the function which multiplies its input by 5 then subtracts 7 from the result. To undo this, one adds 7 to the output, then divides the result by 5. Therefore, the inverse of f is the function
f
−
1
:
R
→
R
{\displaystyle f^{-1}\colon \mathbb {R} \to \mathbb {R} }
defined by
f
−
1
(
y
)
=
y
+
7
5
.
{\displaystyle f^{-1}(y)={\frac {y+7}{5}}.}
Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).
Possible passages
Found by search and not yet checked by a person; they may only mention the idea in passing.
Taking any one of the natural numbers, say 29, it is easy to see that it must have all inductive properties. For since such properties belong to 0 and are hereditary, they belong to 1; therefore, since they are hereditary, they belong to 2, and so on; by twenty-nine repetitions of such arguments we show that they belong to 29. We may _define_ the "inductive" numbers as _all those that possess all inductive…
Bertrand Russell, Our Knowledge of the External World as a Field for Scientific Method in Philosophy
States what Inverse function is.
Thus there arises a classification of modes of formation, or in the language commonly used by mathematicians, a classification of Functions. Any number, considered as formed from any other number, is called a function of it; and there are as many kinds of functions as there are modes of formation. The simple functions are by no means numerous, most functions being formed by the combination of several of the…
John Stuart Mill, A System of Logic, Ratiocinative and Inductive
States what Inverse function is.
There are nevertheless, in mathematics, some examples of so-called Induction, in which the conclusion does bear the appearance of a generalization grounded on some of the particular cases included in it. A mathematician, when he has calculated a sufficient number of the terms of an algebraical or arithmetical series to have ascertained what is called the _law_ of the series, does not hesitate to fill up any number…
John Stuart Mill, A System of Logic
States what Inverse function is.
Related ideas
- Integer
- If and only if
- Function (mathematics)
- Mathematics
- Constructivism (philosophy of mathematics)
- Set (mathematics)
- Set theory
- Subset
- Morphism
- Bijection
- Calculus
- Category theory
- Continuous function
- Axiom of choice
- Function composition
- Categories for the Working Mathematician
Place in the tradition
Part of Outline of logic.