1. Chapter 3 Functions Functions provide a means of expressing relationships between variables, which can be numbers or non-numerical objects.

2. Real Functions They relate two variables x and y which are real numbers. Polynomial, trig, exponential, logarithm, etc. Usually given by formulas y = f(x): y = cos (x) *Even functional relationships that are simple can lead to formulas that are fairly complex. See example of the fuel tank in the text. (p.67) For this reason, we need to study qualitative features of the functional relationship that may not be apparent from the formula.

3. Unit 3.1 “What is a function?” A function is a rule that assigns to each element of a set A a unique element of a set B. (A = B is possible, of course). f is then called ‘a function from A to B’ : a rule or process that tells how to pick the element b in B to be associated with a in A.

4. Function Notation When the function f associates a with b we write f(a) = b called f(x) “f of x” notation, or f:a→b called ‘arrow’ or ‘mapping’ notation. When arrow notation is used we often say that ‘f maps the element a onto b’ and f is called a mapping or map from A to B.

5. Domain, Codomain, Range If f is a function from A to B (f:A → B), the set A is called the domain of f, the set B the codomain of f. The range of f is the subset of B consisting of those elements of B that are actually associated with some element of A by f. We say that f maps A onto the codomain B if every element of B is in the range.

6. Independent, Dependent Variable A value in the domain of a function is called an argument of the function. The variable representing the argument is called the independent variable. The variable that represents the values of the function is called the dependent variable. These are sometimes called the input and output variables.

7. More Vocabulary When f associates b in B with a in A, the element b is called the image of a under f, or the value of f at a. The element a is called the preimage of b under f.

8. Specifying Functions By a formula: y = 2x – 5 By a verbal description of the rule of correspondence: ‘Associate the nth prime number with the natural number n.’ To be able to include all types of correspondences, we need a more precise definition of function. The one used is stated in terms of ordered pairs.

9. Cartesian Product The Cartesian product of two sets A and B, denoted by A x B, is the set of all ordered pairs whose first components are from A and whose second components are from B.

10. Formal Definition of Function For any sets A and B, a function f from A to B is a subset of A x B such that every a in A appears once and only once as the first element of an ordered pair in f. The ordered pair definition is particularly useful for real functions because we can picture the ordered pairs in a graph.

11. Sequences A sequence is a function whose domain is all integers greater than or equal to a fixed integer k (k = 0 or 1 usually). The image of an integer n in a sequence S is called the nth term of the sequence and is usually denoted by rather than s(n). The sequence itself is denoted by S or { } We often only list the range elements.

12. Recursive Definitions Sequences possess a fundamental property that distinguishes them from other types of functions: the possibility of being defined recursively. Example: The Fibonacci sequence