The output of a function is unambiguous if there is only one output for any input. It is especially important to check this when the input of the function can be written in different ways. For example, given by , , and is not unambiguous, because so it is not clear if or .
When giving a function, you must indicate the domain, the codomain, and an unambiguous rule giving an output for every input. There are many, many ways to do this, as long as the output is clear. We gave a few examples during lecture:
We can define the output of the function directly:
We can draw a function:
We might also write this asis given by , , and .
Another way to draw a function is with a table:
This almost describes a function; the domain is clearly (because if there were any other domain elements, would not be a function); and the rule is clear. However, the codomain is not clear: is it ? Or ? Or , or ? When describing a function with a table, the codomain should be specified somewhere.
A function need not have an algorithm for constructing the output; a description of the output is also fine. For example, letgive, on input , the closing Dow-Jones Industrial average on day .