Functions

This page is a sub-page of our page on Mathematical Concepts.

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Sub-pages of this KMR-page:

Continuity
Fourier Series
Fourier Transforms

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Related KMR-pages:

The concept of Function
Basic properties of functions
Relations
Differentiation and Affine Approximation (in One Real Variable)
The natural exponential function \, e^x \,
Uncertainty

In Swedish:

Förändringskalkyl
Osäkerhetskalkyl

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Other relevant sources of information:

Function at Wikipedia
• The problem with functions (by Norman Wildberger):
• Reconsidering functions on modern mathematics (by Norman Wildberger):
• Definitions, specification and interpretation (by Norman Wildberger):

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The interactive simulations on this page can be navigated with the Free Viewer of the Graphing Calculator.

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\, [ \, f_{unction} \, ]_{(D_{omain}, \, C_{odomain} )} \, = \, \left< \, D_{omain} \, \xrightarrow{\,\, f_{unction} \,\,} \, C_{odomain} \, \right>_{(D_{omain}, \, C_{odomain} )}

Definition: \, {C_{odomain}}^{\, D_{omain}} \, \stackrel{\mathrm {def}}{=} \, \{ \, f_{unctions} \, : D_{omain} \, \rightarrow \, C_{odomain} \, \} \,

Applying this definition, we have for two sets \, A \, and \, B \, :

\, {B}^{\, A} \, \equiv \, \{ \, f : A \, \rightarrow \, B \, \} \,

The function \, f : A \, \rightarrow \, B \, induces a function \, f^{-1} : {\mathbf{2}}^{A} \, \leftarrow \, {\mathbf{2}}^{B} \,
from the set of subsets of \, B \, to the set of subsets of \, A .
If \, V \subset B \, the function \, f^{-1} \, is defined by:

\, f^{-1}(V) \, \stackrel{\mathrm {def}}{=} \, \{ \, x \in A \, such that \, f(x) \in V \, \} .

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{f : \mathcal X \, \rightarrow \, \mathcal Y} ,

{\mathcal X \ni x \, \mapsto \, f(x )\in \mathcal Y} ,

\mathcal X \, \stackrel {f} {\longrightarrow} \, \mathcal Y ,

{x \, \longmapsto \, f(x)} ,

{{\mathcal X \, \stackrel {f} {\longrightarrow} \, \mathcal Y \:}\atop {\: x \, \longmapsto \, f(x) } } {\,} .

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A ‘spaghetti’ function
(Ambjörn Naeve on YouTube):

The interactive simulation that created this movie.

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A spaghetti function with coordinate system
(Ambjörn Naeve on YouTube):

The interactive simulation that created this movie.

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Inversion behaves contravariantly under composition of functions, because:

s_{hoe} \circ s_{ock} \, (\, f_{oot} \,) \, \equiv \, s_{hoe} \, (\, s_{ock} \, (\, f_{oot} \,)\,)

(\, s_{hoe} \circ s_{ock} \,)^{-1} (\, f_{oot} \,) \, \equiv \, {s_{ock}}^{-1} ( \, {s_{hoe}}^{-1} (\, f_{oot} \,) \,) \, \equiv \, {s_{ock}}^{-1} \circ {s_{hoe}}^{-1} \, (\, f_{oot} \,)

and hence:

(s_{hoe} \circ s_{ock})^{-1} \, \equiv \, {s_{ock}}^{-1} \circ {s_{hoe}}^{-1} .

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