ABSTRACT
Algebra, just like calculus, works with many different kinds of functions. This chapter aims to learn how to specify functions, and how to compose them. It shows how functions form mathematical structures: semigroups, monoids, and groups. The domain should always be small enough to guarantee that the function rule will work on each element of the domain. The codomain should always be large enough to contain all the function values that occur. In particular, the associativity of matrix multiplication is a direct consequence of the associativity of function composition. Linear functions form one of the most important classes of functions.
