Difference between revisions of "Functional equation"
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===Injectivity and Surjectivity=== | ===Injectivity and Surjectivity=== | ||
− | Consider a function <math>f: X \ | + | Consider a function <math>f: \mathcal{X} \rightarrow \mathcal{Y}</math> be a function <math>f</math> from the set <math>\mathcal{X}</math> to the set <math>\mathcal{Y}</math>, i.e., <math>\mathcal{X}</math> is the domain of <math>f(x)</math> and <math>\mathcal{Y}</math> is the codomain of <math>f(x)</math>. |
− | + | ||
+ | |||
+ | The function <math>f(x)</math> is injective (or one-to-one) if for all <math>a, b</math> in the domain <math>\mathcal{X}</math>, <math>f(a)=f(b)</math> if and only if <math>a=b</math>. Symbolically, | ||
+ | \begin{equation} | ||
+ | f(x) \ \text{is injective} \iff (\forall a,b \in \mathcal{X}, f(a)=f(b)\implies a=b). | ||
+ | \end{equation} | ||
+ | |||
+ | |||
+ | The function <math>f(x)</math> is surjective (or onto) if for all <math>a</math> in the codomain <math>\mathcal{Y}</math> there exists a <math>b</math> in the domain <math>X</math> such that <math>f(b)=a</math>. Symbolically, | ||
+ | \begin{equation} | ||
+ | f(x) \ \text{is surjective} \iff \forall a \in \mathcal{Y},\exists b \in \mathcal{X}: f(b)=a. | ||
+ | \end{equation} | ||
+ | |||
+ | |||
+ | The function <math>f(x)</math> is bijective (or one-to-one and onto) if it is both injective and subjective. Symbolically, | ||
+ | \begin{equation} | ||
+ | f(x) \ \text{is bijective} \iff \forall a \in \mathcal{Y},\exists! b \in \mathcal{X}: f(b)=a. | ||
+ | \end{equation} | ||
+ | |||
+ | The function <math>f(x)</math> has an inverse function <math>f^{-1}(x)</math>, where <math>f^{-1}(f(x)) = x</math>, if and only if it is a bijective function. | ||
==See Also== | ==See Also== |
Revision as of 16:50, 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 be a function from the set to the set , i.e., is the domain of and is the codomain of .
The function is injective (or one-to-one) if for all in the domain , if and only if . Symbolically,
The function is surjective (or onto) if for all in the codomain there exists a in the domain such that . Symbolically,
The function is bijective (or one-to-one and onto) if it is both injective and subjective. Symbolically,
The function has an inverse function , where , if and only if it is a bijective function.