> For the complete documentation index, see [llms.txt](https://mandober.gitbook.io/math-debrief/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://mandober.gitbook.io/math-debrief/260-function-theory/_articles/_basics.md). # Function $$X\times Y = {\forall xy \to x\in X\land y\in Y\ \to (x,y=f(x))}$$ $$\forall x \in X \to \exists y \in Y \to (x,y) \in f$$ $$\forall x \in X \to \exists y \in Y \to (x,y) \in f$$ $$\forall x \in X \to \exists y \in Y \to f(x)=y$$ A function is a relation (process) that associates each element x of a set X, the domain of the function, to a single element y of another set Y (possibly the same set), the codomain of the function. If the function is called f, this relation is denoted y = f (x) (read f of x), the element x is the argument or input of the function, and y is the value of the function, the output, or the image of x by f. The symbol that is used for representing the input is the variable of the function (one often says that f is a function of the variable x). A function is uniquely represented by the set of all pairs (x, f (x)), called the graph of the function. When the domain and the codomain are sets of numbers, each such pair may be considered as the *Cartesian coordinates* of a point in the plane. In general, these points form a curve, which is also called the graph of the function. This is a useful representation of the function, which is commonly used everywhere. ## Links and References ) --- # Agent Instructions This documentation is published with GitBook. GitBook is the documentation platform designed so that both humans and AI agents can read, navigate, and reason over technical content effectively. Learn more at gitbook.com. ## Querying This Documentation If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question. Perform an HTTP GET request on the current page URL with the `ask` query parameter: ``` GET https://mandober.gitbook.io/math-debrief/260-function-theory/_articles/_basics.md?ask= ``` The question should be specific, self-contained, and written in natural language. The response will contain a direct answer to the question and relevant excerpts and sources from the documentation. Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.