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The Rule of Consistency

Module by: Kenny M. Felder. E-mail the author

Summary: This module discusses how functions must be consistent.

There is only one limitation on what a function can do: a function must be consistent.

For instance, the function in the above drawing is given a 5, and gives back a 16. That means this particular function turns 5 into 16—always. That particular function can never take in a 5 and give back a 14. This “rule of consistency” is a very important constraint on the nature of functions.

Note:

This rule does not treat the inputs and outputs the same!

For instance, consider the function y = x 2 y = x 2 . This function takes both 3 and -3 and turns them into 9 (two different inputs, same output). That is allowed. However, it is not reversible! If you take a 9 and turn it into both a 3 and a –3 (two different outputs, same input), you are not a function.

Table 1: If 3 goes in, 9 comes out. If –3 goes in, 9 also comes out. No problem: x2x2 is a function.
3 → − 3 → 3 → − 3 → x squared Gearbox → 9 → 9
Table 2: If 9 goes in, both –3 and 3 come out. This violates the rule of consistency: no function can do this
→ 9 → 9 No function Gearbox 3 → − 3 → 3 → − 3 →

This asymmetry has the potential to cause a great deal of confusion, but it is a very important aspect of functions.

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