Difference between revisions of "Functional equation"
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A classic example of such a function is <math>f(x) = 1/x</math> because <math>f(f(x)) = f(1/x) = x</math>. Cyclic functions can significantly help in solving functional identities. Consider this problem: | A classic example of such a function is <math>f(x) = 1/x</math> because <math>f(f(x)) = f(1/x) = x</math>. Cyclic functions can significantly help in solving functional identities. Consider this problem: | ||
− | Find <math>f(x)</math> such that <math>3f(x) - 4f(1/x) = x^2</math>. | + | Find <math>f(x)</math> such that <math>3f(x) - 4f(1/x) = x^2</math>. Let <math>x=y</math> and <math>x = 1/y</math> in this functional equation. This yields two new equations: |
<math>3f(y) - 4f\left(\frac1y\right) = y^2</math> | <math>3f(y) - 4f\left(\frac1y\right) = y^2</math> |
Revision as of 16:17, 29 August 2024
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
[hide]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: . 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 . Let and in this functional equation. 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,