© 2008 Pearson Addison-Wesley. All rights reserved 8-4-1 Chapter 1 Section 8-4 An Introduction to Functions: Linear Functions, Applications, and Models.

© 2008 Pearson Addison-Wesley. All rights reserved

8-4-1

Chapter 1

Section 8-4An Introduction to Functions: Linear Functions, Applications, and Models


8-4-2

An Introduction to Functions: Linear Functions, Applications, and Models

• Relations and Functions• Domain and Range• Graphs of Relations• Graphs of Functions• Function Notation• Linear Functions• Modeling with Linear Functions


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Terminology

If the value of the variable y depends on the value of the variable x, the y is the dependent variable and x the independent variable.

(x, y)

Independent variable

Dependent variable


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Relation

A relation is a set of ordered pairs.

For example, the sets

F = {(1, 2), (–1, 5), (4, 3)} and

G = {(1, 5), (9, 0), (9, 8)} are both relations.


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Function

A function is a relation in which for each value of the first component of the ordered pairs there is exactly one value of the second component.

Of the two sets,F = {(1, 2), (–1, 5), (4, 3)} and G = {(1, 5), (9, 0), (9, 8)}, only F is a function.


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Example: Determining Independent and Dependent Variables

Determine the independent and dependent variables for the following. The procedure by which someone uses a calculator that finds square roots.

Solution

The independent variable (input) is a nonnegative real number. The dependent variable (output) is the nonnegative square root. For example (81, 9) belongs to this function.


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Domain and Range

In a relation, the set of all values of the independent variable (x) is the domain. The set of all values of the dependent variable (y) is the range.


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Example: Determining Domain and Range

Give the domain and range of the square root function from the previous slide.

Solution

The domain is restricted to the nonnegative numbers:

The range also is

0, .

0, .


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Graphs of Relations

The graph of a relation is the graph of its ordered pairs. The graph gives a picture of the relation, which can be used to determine its domain and range.


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y

x

y

x

Domain {-3, -2, 0, 1}

Range {-2, -1, 2, 3}

Domain [-1, 1]

Range [-3, 3]


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y

x

y

x

Domain

Range

,

,

Domain

Range

,

3,


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Agreement on Domain

The domain of a relation is assumed to be all real numbers that produce real numbers when substituted for the independent variable.


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Graphs of Functions

In a function each value of x leads to only one value of y, so any vertical line drawn through the graph of a function must intersect the graph in at most one point. This is called the vertical line test for a function.


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Vertical Line Test

If a vertical line intersects the graph of a relation in more than one point, then the relation is not a function.


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Example: Vertical Line Test

y

x

y

x

Not a function – the same x-value corresponds to multiple y-values

Function – each x-value corresponds to only one y-value


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Example: Determining Whether a Relation is a Function

Determine whether each equation defines a function and give the domain.

a) 1y x 2b) y x

c) 2y x 3d)

2y

x

Solution

a) Yes, to find the domain note that The domain is

1 0.x 1, .


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Example: Determining Whether a Relation is a Function

Solution (continued)

b) No, (16, 4) and (16, –4) both work. The domain is 0, .

c) No, if x = 1, infinitely many y values satisfy it. The domain is

, .

d) Yes, the domain is since we avoid 0 in the denominator.

, 2 2,


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Variations of the Definition of Function

1. A function is a relation in which for each value of the first component of the ordered pairs there is exactly one value of the

second component.

2. A function is a set of distinct ordered pairs in which no first component is repeated.

3. A function is a rule or correspondence that assigns exactly one range value to each domain value.


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Function Notation

When a function f is defined with a rule or an equation using x and y for the independent and dependent variables, we say “y is a function of x” to emphasize that y depends on x. We use the notation

y = f (x),

called function notation, to express this and read f (x) as “f of x.” For example if y = 3x + 1, we write f (x) = 3x + 1.


8-4-20

Function Notation

Note that f (x) is just another name for the dependent variable y. If f (x) = 3x + 1, we find f (2) by replacing x with 2,

f (2) = 3(2) + 1 = 7

Read f (2) as “f of 2” or “f at 2.”


8-4-21

Example: Using Function Notation

Let f (x) = x 2 + 3x – 1. Find the following.

a) f (2) b) f (0) c) f (2x)

Solutiona) f (2) = 2

2 + 3(2) – 1 = 9

b) f (0) = 0 + 0 – 1 = –1

c) f (2x) = (2x) 2 + 3(2x) – 1 = 4x

2 + 6x – 1


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Linear Functions

A function that can be written in the form

f (x) = mx + b

for real numbers m and b is a linear function.


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Example: Graphing Linear Functions

y

x

y

x

Graph each linear function.a) f (x) = –2x + 1 b) f (x) = 2

Solution


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Example: Modeling with Linear Functions

A company produces DVDs of live concerts. The company pays $200 for advertising the DVDs. Each DVD costs $12 to produce and the company charges $20 per disk.

a) Express the cost C as a function of x, the number of DVDs produced.b) Express the revenue R as a function of x, the number of DVDs sold.c) When will the company break-even? That is, for what value of x does revenue equal cost?


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a) The fixed cost is $200 and for each DVD produced, the variable cost is $12. The cost C can be expressed as a function of x, the number of DVDs produced: C(x) = 12x + 200.

b) Each DVD sells for $20, so revenue R is given by: R(x) = 20x.

Solution


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c) The company will just break even (no profit and no loss) as long as revenue just equals cost, or C(x) = R(x). This is true whenever

12x + 200 = 20x 200 = 8x

25 = x. If 25 DVDs are produced and sold, the company will break even.

Solution (continued)

© 2008 Pearson Addison-Wesley. All rights reserved 8-4-1 Chapter 1 Section 8-4 An Introduction to Functions: Linear Functions, Applications, and Models.

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