Respuesta :
In order to define a function, you need two sets, say [tex]A[/tex] and [tex]B[/tex]. [tex]A[/tex] will be the domain, i.e. the "starting" set. [tex]B[/tex] will be the range (or co-domain), and you can think of this as the "landing" set.
A function is a rule that maps, following the same logic, every element of [tex]A[/tex] into one and only one elment of [tex]B[/tex].
Most of the time, you'll deal with numeric function, i.e. functions that take a number as input and give another number as output. For example, writing
[tex]f(x)=2x[/tex]
means that your function takes a certain number [tex]x[/tex] as input, and return twice that number, [tex]2x[/tex], as output.
Finally, a function can also be described by explicit pair of inputs and outputs. For example, defining a function [tex]f[/tex] with the table
[tex]\begin{array}{c|c}1&2\\4&6\\7&89\end{array}[/tex]
means that [tex]f[/tex] maps 1 onto 2, 4 onto 6 and 7 onto 89. Such a collection of pairs is actually a function if:
- Every point in the domain is actually associated with an output in the range (1)
- Every point in the domain is associated with only one output in the range (2)
(1) for example, if your domain is [tex]A=\{1,2,3,4\}[/tex], and the range is [tex]B=\{13, 14, 55, 59\}[/tex], the table
[tex]\begin{array}{c|c}1&13\\2&14\\3&55\end{array}[/tex]
does not represent a function from [tex]A[/tex] to [tex]B[/tex], because 4 is in the domain, but it is not mapped into anything. Similarly, the table
[tex]\begin{array}{c|c}1&13\\2&14\\3&55\\4&89\end{array}[/tex]
is not a valid function, because 4 is mapped onto 89, which is not in the range.
(2) for example, using the same sets as above, the table
[tex]\begin{array}{c|c}1&13\\1&14\\2&14\\3&55\\4&59\end{array}[/tex]
does not represent a function from [tex]A[/tex] to [tex]B[/tex], because 1 is mapped into both 13 and 14, but a value in the domain can be mapped into only one value in the range.
