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- https://math.libretexts.org/Courses/Monroe_Community_College/MTH_220_Discrete_Math/5%3A_Functions/5.2%3A_De%EF%AC%81nition_of_FunctionsA function \(f\) from a set \(A\) to a set \(B\) (called the domain and the codomain, respectively) is a rule that describes how a value in the codomain \(B\) is assigned to an element from the domain...A function \(f\) from a set \(A\) to a set \(B\) (called the domain and the codomain, respectively) is a rule that describes how a value in the codomain \(B\) is assigned to an element from the domain \(A\).
- https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Elementary_Abstract_Algebra_(Clark)/02%3A_Appendices/2.02%3A_FunctionsHere we collect a few basic facts about functions. Note that the words function, map, mapping and transformation may be used interchangeably. Here we just use the term function. We leave the proofs of...Here we collect a few basic facts about functions. Note that the words function, map, mapping and transformation may be used interchangeably. Here we just use the term function. We leave the proofs of all the results in this appendix to the interested reader.
- https://math.libretexts.org/Courses/SUNY_Schenectady_County_Community_College/Discrete_Structures/06%3A_Functions/6.01%3A_Introduction_to_FunctionsOne of the most important concepts in modern mathematics is that of a function. We often consider a function as some sort of input-output rule that assigns exactly one output to each input. So in this...One of the most important concepts in modern mathematics is that of a function. We often consider a function as some sort of input-output rule that assigns exactly one output to each input. So in this context, a function can be thought of as a procedure for associating with each element of some set, called the domain of the function, exactly one element of another set, called the codomain of the function. This procedure can be considered an input-output-rule.
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Discrete_Mathematics_for_Computer_Science_(Fitch)/03%3A_Functions/3.05%3A_Proof_by_ContradictionThis page discusses direct proof and proof by contradiction, featuring a theorem that the composition of two functions is a function. The proof is divided into two parts: confirming that every domain ...This page discusses direct proof and proof by contradiction, featuring a theorem that the composition of two functions is a function. The proof is divided into two parts: confirming that every domain element maps to a codomain element, and using contradiction to establish uniqueness in mapping. It also introduces mapping composition and explains its operation with two mappings.
- https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/First-Semester_Abstract_Algebra%3A_A_Structural_Approach_(Sklar)/01%3A_Preliminaries/1.02%3A_FunctionsYou have probably encountered functions before. In introductory calculus, for instance, you typically deal with functions from real numbers to real numbers (e.g., the function f(x) = x^2). More gene...You have probably encountered functions before. In introductory calculus, for instance, you typically deal with functions from real numbers to real numbers (e.g., the function f(x) = x^2). More generally, functions “send” elements of one set to elements of another set; these sets may or may not be sets of real numbers. We provide below a “good enough for government work” definition of a function.
- https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Gentle_Introduction_to_the_Art_of_Mathematics_(Fields)/06%3A_Relations_and_Functions/6.05%3A_FunctionsThe concept of a function is one of the most useful abstractions in mathematics. In fact, it is an abstraction that can be further abstracted! For instance, an operator is an entity which takes functi...The concept of a function is one of the most useful abstractions in mathematics. In fact, it is an abstraction that can be further abstracted! For instance, an operator is an entity which takes functions as inputs and produces functions as outputs, thus an operator is to functions as functions themselves are to numbers. There are many operators that you have certainly encountered already – just not by that name.
- https://math.libretexts.org/Bookshelves/Analysis/Introduction_to_Mathematical_Analysis_I_(Lafferriere_Lafferriere_and_Nguyen)/01%3A_Tools_for_Analysis/1.02%3A_FunctionsA function is a collection of ordered pairs and, thus, corresponds to the geometric interpretation of the graph of a function given in calculus.
- https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Book%3A_Mathematical_Reasoning__Writing_and_Proof_(Sundstrom)/06%3A_Functions/6.01%3A_Introduction_to_FunctionsOne of the most important concepts in modern mathematics is that of a function. We often consider a function as some sort of input-output rule that assigns exactly one output to each input. So in this...One of the most important concepts in modern mathematics is that of a function. We often consider a function as some sort of input-output rule that assigns exactly one output to each input. So in this context, a function can be thought of as a procedure for associating with each element of some set, called the domain of the function, exactly one element of another set, called the codomain of the function. This procedure can be considered an input-output-rule.
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Interactive_Linear_Algebra_(Margalit_and_Rabinoff)/03%3A_Linear_Transformations_and_Matrix_Algebra/3.01%3A_Matrix_TransformationsThis page provides an overview of matrix transformations in linear algebra, emphasizing their geometric interpretation in \(\mathbb{R}^2\) and their applications in robotics and computer graphics. It ...This page provides an overview of matrix transformations in linear algebra, emphasizing their geometric interpretation in \(\mathbb{R}^2\) and their applications in robotics and computer graphics. It discusses key concepts such as domain, codomain, range, and the identity transformation while illustrating various transformations like rotation, shear, and projection.
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/A_Spiral_Workbook_for_Discrete_Mathematics_(Kwong)/06%3A_Functions/6.02%3A_Denition_of_FunctionsA function from A to B is a rule that assigns to every element of A a unique element in B .
- https://math.libretexts.org/Bookshelves/Precalculus/Precalculus_(Tradler_and_Carley)/02%3A_Lines_and_Functions/2.02%3A_Introduction_to_functionsWe now formally introduce the notion of a function.
