Math 409-502Harold P. [email protected] definitionA function is given by a rule or by a formula, for example, f (x) = x2.Formal definitionA function is a set of ordered pairs with the property that no first element appears more thanonce, for example, { (a, a2) : a ∈ R }.Math 409-502 October 13, 2004 — slide #2Aside on groupsIn algebra, a group is a set equipped with an associative binary operation for which there is anidentity element and such that each element has an inverse with respect to the operation.ExamplesThe integers form a group under addition.The number 0 is the identity element, and the additive inverse of n is −n.The non-zero real numbers form a group under multiplication.The number 1 is the identity element, and the multiplicative inverse of a is 1/a.Math 409-502 October 13, 2004 — slide #3Groups of functionsDo the real-valued functions with domain R form a group under addition?Yes: the identity element is the function that is constantly equal to 0, and the inverse of f (x) is− f (x).Do the non-zero real-valued functions with domain R form a group under multiplication? Itdepends on what “non-zero” means.If “non-zero” means “not equal to the function that is constantly equal to 0”, then no: thefunction f (x) = x does not have a multiplicative inverse that is everywhere defined.If “non-zero” means “nowhere equal to zero”, then yes: the function that is constantly equalto 1 is the multiplicative identity, and the inverse of f (x) is 1/ f (x).Math 409-502 October 13, 2004 — slide #4CompositionDo the real-valued functions with domain R form a group under composition?The identity function f (x) = x serves as identity under composition.But some functions lack inverses.The function f (x) = x sin(x) is not one-to-one, so that function does not have an inverse undercomposition.The function g(x) = exis one-to-one but not onto, so its inverse function ln(x) is not everywheredefined.Math 409-502 October 13, 2004 — slide #5Homework1. Read sections 10.1 and 10.2, pages 137–142.2. Do Exercises 10.1/2 and 10.2/1 on page 148.Math 409-502 October 13, 2004 — slide