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Chapter 1
Section 8-4
An Introduction to Functions: Linear
Functions, Applications, and Models
8-4-1
© 2008 Pearson Addison-Wesley. All rights reserved
An Introduction to Functions: Linear
Functions, Applications, and Models
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Relations and Functions
Domain and Range
Graphs of Relations
Graphs of Functions
Function Notation
Linear Functions
Modeling with Linear Functions
8-4-2
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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.
Independent
variable
Dependent
variable
(x, y)
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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.
8-4-5
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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: 0,   .
The range also is 0,   .
8-4-8
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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.
8-4-9
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Example: Determining Domain and
Range
y
y
x
x
Domain {-3, -2, 0, 1}
Domain [-1, 1]
Range {-2, -1, 2, 3}
Range [-3, 3]
8-4-10
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Example: Determining Domain and
Range
y
y
x
Domain  ,  
Range
 ,  
x
Domain  ,  
Range
3, 
8-4-11
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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.
8-4-12
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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.
8-4-13
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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
y
x
Not a function – the same
x-value corresponds to
multiple y-values
x
Function – each x-value
corresponds to only one
y-value
8-4-15
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Example: Determining Whether a
Relation is a Function
Determine whether each equation defines a
function and give the domain.
a) y  x  1
b) y 2  x
c) y  2 x
3
d) y 
x2
Solution
a) Yes, to find the domain note that x  1  0.
The domain is 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  , 2  2,  
since we avoid 0 in the denominator.
8-4-17
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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.
8-4-18
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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.
8-4-19
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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.”
8-4-20
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Example: Using Function Notation
Let f (x) = x 2 + 3x – 1. Find the following.
a) f (2)
b) f (0)
c) f (2x)
Solution
a) 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
8-4-21
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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.
8-4-22
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Example: Graphing Linear Functions
Graph each linear function.
a) f (x) = –2x + 1
b) f (x) = 2
Solution
y
y
x
x
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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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Example: Modeling with Linear Functions
Solution
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.
8-4-25
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Example: Modeling with Linear Functions
Solution (continued)
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.
8-4-26
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