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 subtract the second equation from the first, we have: |
<math>25f(y) = 3y^2 - \frac{4}{y^2}</math> | <math>25f(y) = 3y^2 - \frac{4}{y^2}</math> | ||
− | So clearly, <math>f(y) = \frac{3}{25}y^2 - \frac{4}{25y^2}</math> | + | So, clearly, <math>f(y) = \frac{3}{25}y^2 - \frac{4}{25y^2}</math> |
==See Also== | ==See Also== | ||
*[[Functions]] | *[[Functions]] | ||
*[[Polynomials]] | *[[Polynomials]] |
Revision as of 18:06, 11 July 2006
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,