Difference between revisions of "Functional equation"
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Revision as of 12:52, 23 June 2006
Functional Equations are equations that involve functions. For an example, these are some examples of functional equations:
Contents
Introductory Topics
The Inverse of a Function
The inverse of a function is a function that "undoes" a function. For an example, consider the function: f(x). The function has the property that . In this case is called the inverse function. Often the inverse of a function is denoted by .
Intermediate Topics
Cyclic Functions
A cycylic function is a function that has the property that:
A classic example of such a function is because . Cyclic functions can significantly help in solving functional identities. Consider this problem:
Find such that . In this functional equation, let and let . This yields two new equations:
Now, if we multiply the first equation by 3 and the second equation by 4, and substract the second equation from the first, we have:
So clearly,