# Functions

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Sub-pages of this KMR-page:

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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:

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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

The interactive simulation that created this movie.

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A spaghetti function with coordinate system

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:

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