Chapter 3 Functions

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 aunique 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 independentvariable. • The variable that represents the values of the function is called the dependentvariable. • These are sometimes called the input and outputvariables.

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 orderedpairs.

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