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The following are all examples of functions:

1. \(f:\Z \to \Z\) defined by \(f(n) = 3n\text{.}\) The domain and codomain are both the set of integers. However, the range is only the set of integer multiples of 3.

2. \(g: \{1,2,3\} \to \{a,b,c\}\) defined by \(g(1) = c\text{,}\) \(g(2) = a\) and \(g(3) = a\text{.}\) The domain is the set \(\{1,2,3\}\text{,}\) the codomain is the set \(\{a,b,c\}\) and the range is the set \(\{a,c\}\text{.}\) Note that \(g(2)\) and \(g(3)\) are the same element of the codomain. This is okay since each element in the domain still has only one output.

3. \(h:\{1,2,3,4\} \to \N\) defined by the table:

   \(x\)	1	2	3	4
   \(h(x)\)	3	6	9	12

   Here the domain is the finite set \(\{1,2,3,4\}\) and to codomain is the set of natural numbers, \(\N\text{.}\) At first you might think this function is the same as \(f\) defined above. It is absolutely not. Even though the rule is the same, the domain and codomain are different, so these are two different functions.