FUNCTIONS AND MODELSCHAPTER 1, SECTION 1

Function Definitions

A function is a rule or correspondence that assigns to each element of one set exactly one element of a second set.

The set of elements in the first set is called the domain. If x is any element in the domain, then it is called the independent variable.

The set of elements in the second set is called the range. If y is an output of the function from an input x, then y is called the dependent variable.

Functions may be defined as a set of ordered pairs, a table, a graph, an equation or a verbal statement.

Example: Find the domain.

a)

b)

c)

To make sure that the function results in a real number value, watch for two types of exceptions:

1. Values from the domain that make the denominator 0 (zero).

2. Values in the domain that make a negative value under an even root.

Recognizing Functions

Is the table a function?

Does the equation

define y as a function of x?

Does the equation

define y as a function of x?

x Year N Worldwide PCs(millions)1991 129.41992 150.81993 177.41994 2081995 2452000 535.62005 903.9

For ordered pairs, if the independent variable values are all different, the table is a function.

For equations, if the power on the dependent variable(y) is odd, then the equations defines the dependent variable(y) as a function of the independent variable(x).

In the case of graphs, the Vertical Line Test is the way to determine if the graph represents a function.

Vertical Line Test: a graph is a function if no vertical line intersects the graph in more than one point.

a)

b)

c)

Function Notation

We can use the function notation , read “y equals f of x” to indicate that the variable y is a function of the variable x.

For specific values of x, f(x) represents the resulting y values especially when referring to a graph generated by the function. The point (a, f(a)) lies on the graph for any given a in the domain of the function.

For example: represents a function and

for

Mathematical Models

The process of transforming real-world information into a mathematical form that can be applied and interpreted is called modeling.

For this reason, a mathematical model is a functional relationship that includes the function and a description of all variables and their units of measure.

Example: WeightDuring the first two weeks in May, a man weighs himself daily(in pounds) and records the data. a. Does the table define

weight as a function of the day in May?

b. What is the domain?c. What is the range?d. During what day(s)

did he weigh the most?

e. During what day(s) did he weigh the least?

May Weight(lb) May Weight(lb)

1 178 8 176

2 177 9 175

3 178 10 174

4 177 11 173

5 176 12 173

6 176 13 172

7 175 14 171