copyright © 2010 pearson education, inc. publishing as prentice hall. 1.2 basics of functions and...

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

1.2

Basics of Functions and Their Graphs

Blitzer, College Algebra: An Early Functions Approach, 2ed 22


Relations

The test grades of 22 students in Mrs. Smith’s Algebra class are shown in the table below. The table indicates a correspondence between a grade and the number of students receiving that grade.

Grade # of Students

A 5

B 7

C 8

D 1

F 1

We can write this correspondence as a set of ordered pairs:

{(A, 5), (B, 7), (C, 8), (D, 1), (F, 1)}.

The mathematical term for a set of ordered pairs is a relation.



Relations

Example:

Definition of a Relation A relation is any set of ordered pairs. The set of all first components of the ordered pairs is called the domain of the relation and the set of all second components is called the range of the relation.

Definition of a Relation A relation is any set of ordered pairs. The set of all first components of the ordered pairs is called the domain of the relation and the set of all second components is called the range of the relation.

Find the domain and range of the relation:{(A, 5), (B, 7), (C, 8), (D, 1), (F, 1)}.

Domain: {A, B, C, D, F} Range: {5, 7, 8, 1}

The 1 is only listed once.



Functions

Grade # of Students

A 5

B 7

C 8

D 1

F 1

Grade

ABCDF

# of Students

5781

Domain

Grades corresponding to number of students.

Range

A relation in which each member of the domain corresponds to exactly one member of the range is a function.

This relation is a function.



Functions

Grade # of Students

A 5

B 7

C 8

D 1

F 1

# of Students

5781

Grade

ABCDF

DomainNumber of students corresponding to grades.

Range

Definition of a FunctionA function is a correspondence from a first set, called the domain, to a second set, called the range, such that each element in the domain corresponds to exactly one element in the range.

Definition of a FunctionA function is a correspondence from a first set, called the domain, to a second set, called the range, such that each element in the domain corresponds to exactly one element in the range.

This relation is NOT a function.



Functions

Example:

Determine whether the relation is a function:

{(3, 2), (0, 8), (2, 3), (3, 4), (4, 4)}.

3024

2834

Domain Range

This is not a function because 3 corresponds to both 2 and 4.

This is not a function because 3 corresponds to both 2 and 4.

A function can have two different first components with the same second component. By contrast, a relation is not a function when two different ordered pairs have the same first component and different second components



Functions as Equations

Functions are usually given in terms of equations rather than as sets of ordered pairs.

y = 3x + 5

For each value of x, there is only one value of y.

y is a function of x.

The variable x is called the independent variable because it can be assigned any value from the domain.

The variable y is called the dependent variable because its value depends on x.



Functions as Equations

Example:

Determine whether the equation defines y as a function of x.

x2 + y2 = 9

Solve the equation for y in terms of xx2 + y2 = 9

x2 + y2 – x2 = 9 – x2 Solve for y.

y2 = 9 – x2 Simplify.

Since two or more values of y can be obtained for a given x, the equation is not a function.

29y x Apply the square root property.



Function Notation

If an equation in x and y gives one and only one value of y for each value of x then the variable y is a function of the variable x.

The special notation f(x), read “f of x” or “f at x,” represents the value of the function at the number x.

x f(x) f(x) = 3x2 + 8

Input Output Equation This is read as f(x) = 3x2 + 8.



Evaluating a Function

Example:

Evaluate the function for f(– 2).

f(x) = 3x2 + 8

f(x) = 3(– 2)2 + 8 Substitute – 2 for x in the equation.

= 3(4) + 8 Square – 2.

= 12+ 8 Multiply.

= 12+ 8 Add.

= 20 Simplify.



Graphs of Functions

Example:Graph the function f(x) = x + 3.

x f(x) = – x + 3 (x, y)

1 y = (1) + 3 = 4 (1, 4)

2 y = (2) + 3 = 5 (2, 5)

1 y = (– 1) + 3 = 2 (1, 2)

(1, 4)

x

y

1234 1 2 3 4

2

34

1

2

3

4

(1, 2)

(2, 5)

f(x) = – x + 3f(x) = – x + 3

The graph of a function is the graph of its ordered pairs.



The Vertical Line Test

The Vertical Line Test for Functions If any vertical line intersects a graph in more than one point, the graph does not define y as a function of x.

The Vertical Line Test for Functions If any vertical line intersects a graph in more than one point, the graph does not define y as a function of x.

FUNCTIONNOT A

FUNCTIONFUNCTION

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