Graph of a function: Difference between revisions

Revision as of 03:44, 6 October 2020

Revision as of 03:44, 6 October 2020

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</math> is a subset of <math>\mathbb R^{n+m}</math> (strictly speaking it is <math>\mathbb R^n \times \mathbb R^m</math>, but one can embed it with the natural isomorphism).

== Examples ==

1p9qzw02ukraph <math>\{1,2,3\}=\{x:\ \text{there exists } y,\text{ such that }(x,y)\in G(f)\}</math>.

=== Functions of one variable ===

[[File:Three-dimensional graph.png|right|thumb|250px|Graph of the [[function (mathematics)|function]] {{nowrap|1=''f''(''x'', ''y'') = sin(''x''²) · cos(''y''²)}}.]]

The graph of the function <math>f:\{1,2,3\}\to \{a,b,c,d\}</math> defined by

: <math>f(x)=

\begin{cases}

a, & \text{if }x=1, \\ d, & \text{if }x=2, \\ c, & \text{if }x=3,

\end{cases}

</math>

is the subset of the set <math>\{1,2,3\}\times \{a,b,c,d\}</math>

: <math>G(f) = \{ (1,a), (2,d), (3,c) \}. \, </math>

1p9qzw02ukraphFrom the graph, the ___domain <math>\{1,2,3\}</math> is recovered as the set of first component of each pair in the graph <math>\{1,2,3\}=\{x:\ \text{there exists } y,\text{ such that }(x,y)\in G(f)\}</math>.

Similarly, the [[Range of a function|range]] can be recovered as <math>\{a,c,d\}=\{y: \text{there exists }x,\text{ such that }(x,y)\in G(f)\}</math>.

The codomain <math>\{a,b,c,d\}</math>, however, cannot be determined from the graph alone.