Difference between revisions of "Functional equation"
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<math>3f\left(\frac1y\right)- 4f(y) = \frac1{y^2}</math> | <math>3f\left(\frac1y\right)- 4f(y) = \frac1{y^2}</math> | ||
| − | Now, if we multiply the first equation by 3 and the second equation by 4, and | + | Now, if we multiply the first equation by 3 and the second equation by 4, and add the two equations, we have: |
| − | <math> | + | <math>-7f(y) = 3y^2 + \frac{4}{y^2}</math> |
| − | So, clearly, <math>f(y) = \frac{3}{ | + | So, clearly, <math>f(y) = -\frac{3}{7}y^2 - \frac{4}{7y^2}</math> |
=== Problem Examples === | === Problem Examples === | ||
Revision as of 08:45, 4 February 2009
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 add the two equations, we have:
So, clearly, 
