Difference between revisions of "Functional equation"
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=== Problem Examples === | === Problem Examples === | ||
* [[2006_AMC_12A_Problems/Problem_18 | 2006 AMC 12A Problem 18]] | * [[2006_AMC_12A_Problems/Problem_18 | 2006 AMC 12A Problem 18]] | ||
| − | + | * [[2007_AIME_II_Problems/Problem_14 | 2007 AIME II Problem 14]] | |
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==See Also== | ==See Also== | ||
Revision as of 23:46, 24 November 2007
A functional equation, roughly speaking, is an equation in which some of the unknowns to be solved for are functions. For example, the following are 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 (right) inverse function. (Similarly, a function so that 
is called the left inverse function. Typically the right and left inverses coincide on a suitable domain, and in this case we simply call the right and left inverse function the inverse function.) Often the inverse of a function 
is denoted by 
.
Intermediate Topics
Cyclic Functions
A cyclic 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 subtract the second equation from the first, we have:
So, clearly, 
