Difference between revisions of "Functional equation"
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===Injectivity and Surjectivity=== | ===Injectivity and Surjectivity=== | ||
+ | Consider a function <math>f: X \arrow Y</math>. | ||
+ | |||
+ | An function <math>f(x)</math> is injective if for all <math>a, b</math> in the domain of <math>f(x)</math>, <math>f(a)=f(b)</math> if and only if <math>a=b</math> | ||
==See Also== | ==See Also== |
Revision as of 16:21, 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,
Problem Examples
Advanced Topics
Injectivity and Surjectivity
Consider a function $f: X \arrow Y$ (Error compiling LaTeX. Unknown error_msg).
An function is injective if for all in the domain of , if and only if