Functions

Calculus is the mathematical study of change, and real-life things that change are modeled by functions. In this module we will concern ourselves first with learning about general functions, and later with certain types of common functions with special properties, like trigonometric and polynomial functions. First, it is of critical importance to understand exactly what a function is. We will discuss what makes a function a function, some general properties of functions, and a few basic categories of functions. In this module we will assume a general knowledge of algebraic principles of solving equations, working with the real numbers, and working with sets.

Terminologies

Cartesian Product- The set of all possible ordered pairs (a,b) composed of elements taken from the two sets,AandB.

Composition- A function operation symbolized (fog)(x) that is equivalent to f(g(x)).

Defined- is defined at a given value of the independent variable if it assigns that input an output;defined means "takes on a value".

Dependent Variable- The output variable of a function; the variable whose value depends on the input, or independent variable.

Domain- The set of all inputs for which a function or relation is defined.

Even Function- A function fis even if f(x) =f(-x).

Function- A relation which assigns exactly one element in its range for each element in its domain.

Horizontal Line Test- The test by which it is shown whether a function is a one-to-one function or not, and therefore whether its inverse is a function.

Independent Variable- The variable of a function which does not depend on the other variable -- it is the input.

Inverse- A relation which assigns a correspondence from the elements of the range to those of the domain. The inverse of a function or relation can be found by interchanging the variables in the function or relation.

Odd Function- A function f is odd if f(x) = -f(-x).

One-to-One Function- A function is one-to-one if each element in its range is paired with exactly one element from its domain.

Periodic Function- A function is periodic if and only iff(x) =f(x+c), for all valuesx, wherecis a constant. A periodic function repeats itself at regular intervals.

Piecewise Function- A function is piecewise if and only if it uses different rules for different parts of its domain.

Range- The set of all outputs of a function or relation.

Relation- A rule that associates the elements of one set with those of another set. A relation can also be thought of as all of the ordered pairs which satisfy the rule.

Undefined- A function is undefined at a given value of its independent variable if for that value, there is no output--this occurs when a particular input creates a situation in which there is division by zero, or an even root of a negative number, for example.

Vertical Line Test- The test by which a relation is either shown to be a function or not. The graph of a function does not intersect with a vertical line more than once.

Sets and Relations

Sets

A set is any collection of objects. Some examples of sets could include the following:

1) all the boys in a classroom;

2) all the restaurants in New York;

3) the boxes of golf balls on sale in a given store.

The objects in a set are called elements. The elements of the above sets are boys, restaurants, and boxes of golf balls, respectively. For every set there is some rule that distinguishes the elements in the set from other things.

In math, sets usually either consist of quantities of things, or numbers themselves. In a given problem, two sets might be the scores of a class on one test, and the scores of the same students on another test. Using different mathematical techniques, these sets can be compared extensively. The other sets commonly found in the study of math actually consist of numbers. These are sets like whole numbers, natural numbers, integers, real numbers, etc. They also have a rule that distinguishes their elements from other numbers.

Relations

Two sets can be associated according to a rule. Given two sets,AandB, the set of all the possible ordered pairs in which the first element comes from A and the second element comes from B is called the Cartesian product AB. Given a rule that associates elements of A,a, with elements ofB,b, there may exist ordered pairs (a,b) that satisfy the rule. The set of all ordered pairs (a,b)that satisfy the rule is called a relation. A relation between two sets is a subset of the Cartesian product of those sets. The domain of a relation is the set of all the first elementsa of the ordered pairs. The range of a relation is the set of all the second elements of the ordered pairs, b.

Consider the setsX and Y: X= {1, 2, 3, 4}, and Y= {12, 14, 21}. A relation between X and Y can be defined by the rule y= 7x. The ordered pairs that satisfy this rule compose the relation (x,y). They are(2, 14) and (3, 21). These two ordered pairs form the relation. The domain of the relation is the set D= {2, 3}, and the range is the setR= {14, 21}.

Relations are often special associations between elements of the same set. In this module, most of the relations we will see associate elements of the real numbers with other elements of the real numbers. Relations between a set and itself are not uncommon.

Problems

1. Let set A= {1, 2, 3} and set B= {4, 5}. What is their Cartesian product AB?

Answer:

AB= {(1, 4),(1, 5),(2, 4),(2, 5),(3, 4),(3, 5)}.

This is every possible ordered pair from which the first element is taken from Aand the second element taken from B.

1. What is the Cartesian product of the set of positive real numbers with itself?

Answer:

The Cartesian product is every ordered pair in the first quadrant of the coordinate plane.

1. What is the domain and range of the relation y=