Slide 2.4- 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley.

Slide 2.4- 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

OBJECTIVES

Relations and Functions

Learn the definition of a relation.

Learn the definition of a function and learn how to determine which relations are functions.

Learn to use functional notation and find function values.

Learn to find the domain of a function.

SECTION 2.4

1

2

3

4

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

OBJECTIVES

Relations and Functions

Learn to identify the graph of a function.

Learn to find the average rate of change of a function.

Learn to solve applied problems by using functions.

SECTION 2.4

5

6

7


DefinitionsAn ordered pair (a, b) is said to satisfy an equation with variables a and b if, when a is substituted for x and b is substituted for y in the equation, the resulting statement is true.

An ordered pair that satisfies an equation is called a solution of the equation.

Frequently, the numerical values of the variable y can be determined by assigning appropriate values to the variable x. For this reason, y is sometimes referred to as the dependent variable and x as the independent variable.


DEFINTION OF A RELATION

Any set of ordered pairs is called a relation.The set of all first components is called the domain of the relation, and the set of all second components is called the range of the relation.


EXAMPLE 1 Finding the Domain & Range of a Relation

Find the domain and range of the relation

Solution

The domain is the set of all first components, or {Titanic, Star Wars IV, Shrek 2, E.T., Star Wars I, Spider-Man}.

{ (Titanic, $600.8), (Star Wars IV, $461.0), (Shrek 2, $441.2), (E.T., $435.1), (Star Wars I, $431.1), (Spider-Man, $403.7)}.

The range is the set of all second components, or {$600.8, $461.0, $441.2, $435.1, $431.1, $403.7)}.


DEFINTION OF A FUNCTION

A function from a set X to a set Y is a relation in which each element of X corresponds to one and only one element of Y.


CORRESPONDENCE DIAGRAMA relation may be defined by a correspondence diagram, in which an arrow points from each domain element to the element or elements in the range that correspond to it.


EXAMPLE 2 Determining Whether a Relation is a Function

Determine whether the relations that follow are functions. The domain of each relation is the family consisting of Malcolm (father), Maria (mother), Ellen (daughter), and Duane (son).

a. For the relation defined by the following diagram, the range consists of the ages of the four family members, and each family member corresponds to that family member’s age.





Solution

The relation is a function, because each element in the domain corresponds to exactly one element in the range. For a function, it is permissible for the same range element to correspond to different domain elements. The set of ordered pairs that define this relation is {(Malcolm, 36), (Maria, 32), (Ellen, 11), (Duane, 11)}.



b. For the relation defined by the diagram on the next slide, the range consists of the family’s home phone number, the office phone numbers for both Malcolm and Maria, and the cell phone number for Maria. Each family member corresponds to all phone numbers at which that family member can be reached.





The relation is not a function, because more than one range element corresponds to the same domain element. For example, both an office phone number and a home phone number correspond to Malcolm. The set of ordered pairs that define this relation is {(Malcolm, 220-307-4112), (Malcolm, 220-527-6277 ), (Maria, 220-527-6277), (Maria, 220-416-5204), (Maria, 220-433-8195), (Ellen, 220-527-6277), (Duane, 220-527-6277)}.

Solution


FUNCTION NOTATIONWe usually use single letters such as f, F, g, G, h, H, and so on as the name of a function. For each x in the domain of f, there corresponds a unique y in its range. The number y is denoted by f (x) read as “f of x” or “f at x”. We call f(x) the value of f at the number x and say that f assigns the f (x) value to x. Because the value of y depends on the given value of x, y is called the dependent variable and x is called the independent variable.


FUNCTION NOTATION

We can describe functions in many ways:

A table


FUNCTION NOTATION

A graph


FUNCTION NOTATION

An equation

y x2

f x x2

y x2 6x 8

g x x2 6x 8


EXAMPLE 3 Evaluating a Function

Solution

Let g be the function defined by the equationy = x2 – 6x + 8.Evaluate each function value.

a. g 3 b. g 2 c. g1

2

d. g a 2 e. g x h

a. g 3 32 6 3 8 1

b. g 2 2 2 6 2 8 24


EXAMPLE 3 Evaluating a Function

Solution continued

c. g1

2

1

2

2

61

2

8

21

4

d. g a 2 a 2 2 6 a 2 8

a2 4a 4 6a 12 8

a2 2a

e. g x h x h 2 6 x h 8

x2 2xh h2 6x 6h 8


EXAMPLE 4Determining Whether an Equation Defines a Function

Determine whether each equation determines y as a function of x. a. 6x2 – x – 2 b. y2 – x2 = 4

One value of y corresponds to one value of x so it defines y as a function of x.

Solution6x2 3y 12

6x2 3y 3y 12 12 3y 12

6x2 12 3y

2x2 4 y

a.


EXAMPLE 4Determining Whether an Equation Defines a Function

Solution continued

b. y2 – x2 = 4

Two values of y correspond to the same value of x so it does not define y as a function of x.

y2 x2 4

y2 x2 x2 4 x2

y2 x2 4

y x2 4


AGREEMENT ON DOMAIN

If the domain of a function that is defined by an equation is not explicitly specified, then we take the domain of the function to be the largest set of real numbers that result in real numbers as outputs.


EXAMPLE 5 Finding the Domain of a Function

Find the domain of each function.

a. f x 1

1 x2

Solution

a. f is not defined when the denominator is 0.

b. g x x

c. h x 1

x 1d. P t 2t 1

1 x2 0

x 1Domain: {x|x ≠ –1 and x ≠ 1}

° , 1 U 1,1 U 1, °



Solution continued

The square root of a negative number is not a real number and is excluded from the domain.

b. g x x

c. h x 1

x 1

Domain: {x|x ≥ 0}, [0, ∞)

The square root of a negative number is not a real number and is excluded from the domain, so x – 1 ≥ 0. However, the denominator ≠ 0.



Solution continued

So x – 1 > 0 so x > 1.

d. P t 2t 1

Domain: {x|x > 1}, or (1, ∞)

Any real number substituted for t yields a unique real number.

Domain: {t|t is a real number}, or (–∞, ∞)


EQUALITY OF FUNCTIONS

Two functions f and g are equal if and only if

1. f and g have the same domain and

2. f (x) = g (x) for all x in the domain.


VERTICAL LINE TEST

If no vertical line intersects the graph of a relation at more than one point, then the graph is the graph of a function.


EXAMPLE 6

Use the vertical-line test to determine which graphs are graphs of functions.

Identifying the Graph of a Function

Solution

Not a function Does not pass the vertical line test


EXAMPLE 6



Solution

Not a function Does not pass the vertical line test


EXAMPLE 6



Solution

Is a functionDoes pass the vertical line test


EXAMPLE 6



Solution

Is a functionDoes pass the vertical line test


EXAMPLE 7

Leta. Is the point (1, –3) on the graph of f ?b. Find all values of x such that (x, 5) is on the

graph of f.c. Find all y–intercepts of the graph of f.d. Find all x–intercepts of the graph of f.

Examining the Graph of a Function

y f x x2 2x 3.

Solutiona. Check whether (1, –3) satisfies the equation.

(1, –3) is not on the graph of f.

f x y x2 2x 3

3 1 2 2 1 3 4 No!?


EXAMPLE 7

b. Find all values of x such that (x, 5) is on the graph of f.Substitute 5 for y and solve for x.


Solution continued

5 x2 2x 3

0 x2 2x 8

0 x 4 x 2 x 2 or x 4

(–2, 5) and (4, 5) are on the graph of f.


EXAMPLE 7

c. Find all y–intercepts of the graph of f. Substitute 0 for x and solve for y.


Solution continued

y x2 2x 3

y 02 2 0 3

y 3

The only y-intercept is –3.


EXAMPLE 7

d. Find all x–intercepts of the graph of f.Substitute 0 for y and solve for x.


Solution continued

y x2 2x 3

0 x2 2x 3

0 x 1 x 3 x 1 or x 3

The x–intercepts of the graph of f are –1 and 3.


THE AVERAGE RATE OF CHANGE OF A FUNCTION

f b f a b a

, a b.

Let (a, f (a)) and (b, f (b)) be points on the graph of a function f. Then the average rate of change of f (x) as x changes from a to b is defined by


EXAMPLE 8

Find the average rate of change off (x) = 2x2 – 3 as x changesa. from x = 2 to x = 4;b. from x = 3 to x = 6;c. from x = c to x = c + h, h ≠ 0.

Finding the Average Rate of Change

f 4 f 2

4 2

2 4 2

– 3 2 2 2– 3

4 212

Solution

Average rateof change


EXAMPLE 8

b. from x = 3 to x = 6;


Solution continued

f 6 f 3

6 3

2 6 2

– 3 2 3 2– 3

6 3

69 15

3

54

318

Average rate of change


EXAMPLE 8

c. from x = c to x = c + h, h ≠ 0.


Solution cont’

Average rate of change

f c h f c

c h c

2 c h 2

– 3 2 c 2– 3

c h c

2 c2 2ch h2 – 3 2c2 – 3

h

4ch 2h2

h

h 4c 2h h

4c 2h


DIFFERENCE QUOTIENT

f x h f x h

, h 0,

For a function f , the quantity

is called the difference quotient.


EXAMPLE 9

Let f (x) = 2x2 – 3 + 5. Find and simplify

Evaluating and Simplifying a Difference Quotient

Solutionf x h 2 x h 2 3 x h 5

2 x2 2xh h2 3 x h 5

2x2 4xh 2h2 3x 3h 5

Find

f x h f x h

, h 0,

Now substitute into the difference quotient.


EXAMPLE 9Evaluating and Simplifying a Difference Quotient

Solution continued

2x2 4xh 2h2 3x 3h 5 2x2 3x 5

h

2x2 4xh 2h2 3x 3h 5 2x2 3x 5

h

4xh 3h 2h2

h

h 4x 3 2h h

4x 3 2h


EXAMPLE 10

Many drugs used to lower high blood cholesterol levels are called statins and are very popular and widely prescribed. These drugs, along with proper diet and exercise, help prevent heart attacks and strokes. Recall from the introduction to this section that bioavailability is the amount of a drug you have ingested that makes it into your bloodstream.

Cholesterol–Reducing Drugs


EXAMPLE 10

A statin with a bioavailability of 30% has been prescribed for Boris to treat his cholesterol levels. Boris is to take 20 milligrams of this statin every day. During the same day, one-half of the statin is filtered out of the body. Find the maximum concentration of the statin in the bloodstream on each of the first ten days of using the drug, and graph the result.



EXAMPLE 10

Since the statin has 30% bioavailability and Boris takes 20 milligrams per day, the maximum concentration in the bloodstream is 30% of 20 milligrams, or 20(0.3) = 6 milligrams from each day’s prescription. Because one-half of the statin is filtered out of the body each day, the daily maximum concentration is


Solution

1

2previous day's maximum concentration 6


EXAMPLE 10 Cholesterol–Reducing Drugs

Solution continued

Day Max Concentration

1 6.000

2 9.000

3 10.500

4 11.250

5 11.625

6 11.813

7 11.906

8 11.953

9 11.977

10 11.988Maximum concentration approaches 12 mg.


FUNCTIONS IN ECONOMICS

C x variable cost fixed costs ax b

b is the fixed costa (the cost of producing each item in dollars per item) is called the marginal cost.

Linear Cost Function

C x C x x

Average Cost



p x mx d

or

x p np k

Suppose x items can be sold (demanded) at a price of p dollars per item.

Linear Price–Demand Function



R x pgx mx d x

Revenue Function

Profit Function

Revenue = (Price per item) (Number of items sold)

P x R x C x Profit = Revenue – Cost


EXAMPLE 11

Metro Entertainment Co. spent $100,000 on production costs for its off-Broadway play Bride and Prejudice. Once running, each performance costs $1000 per show and the revenue from each is $2400. Using x to represent the number of shows,

Breaking Even

a. Write cost function C(x).b. Write revenue function R(x).c. Write profit function P(x).d. How many showings of Bride and Prejudice

must be held for Metro to break even?


EXAMPLE 11 Breaking Even

Solution

a. C x Variable cost fixed cost 1000x 100, 000

b. R x Revenue per show Number of shows 2400x

c. P x R x C x 2400x 1000x 100, 000 1400x 100, 000


EXAMPLE 11 Breaking Even

Solution continued

1400x 100,000 0

1400x 100,000

x 100,000

140071.429

d. Metro will break even when P(x) = 0.

So 72 is the number of shows required for Metro to break even.

Slide 2.4- 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley.

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