Chapter 3 Functions

Functions provide a means of expressing relationships between variables, which

can be numbers or non-numerical objects.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

nsns

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