Download Section 8-4

Chapter 1
Chapter 8
Graphs,
Functions, and
Systems of
Equations and
Inequalities
Section 8-4
An Introduction to Functions: Linear
Functions, Applications, and Models
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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
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
Relation
Relations can be defined by equations.
A relation is a set of ordered pairs.
For example, the sets
y = x + 1 is the set of all ordered pairs (x, y) where
the second coordinate (y) is one greater than the
first coordinate (x).
F = {(1, 2), (–1, 5), (4, 3)} and
G = {(1, 5), (9, 0), (9, 8)} are both relations.
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Function
Domain and Range
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.
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.
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 Domain and
Range
Graphs of Relations
y
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.
y
x
x
Domain {-3, -2, 0, 1}
Domain -1 ≤ x ≤ 1
Range {-2, -1, 2, 3}
Range -3 ≤ y ≤ 3
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Example: Determining Domain and
Range
y
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y
x
Interval notation
Domain and range can be expressed using interval
notation.
x
Domain is all x
Domain is all x
Range is all y
Range -3 ≤ y ≤ ∞
Recall that for our ellipse:
Domain -1 ≤ x ≤ 1
Range -3 ≤ y ≤ 3
Using interval notation:
x ∈ [-1, 1], y ∈ [-3, 3]
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Agreement on Domain
Interval notation
[a, b] means all values from a to b, including the
endpoints.
Same as a ≤ x ≤ b
The domain of a relation is assumed to be all
real numbers that produce real numbers when
substituted for the independent variable.
(a, b) means all values from a to b, not including
the endpoints.
Same as a < x < b
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Graphs of Functions
Vertical Line Test
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.
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: Determining Whether a
Relation is a Function
Example: Vertical Line Test
y
y
x
Not a function – the same
x-value corresponds to
multiple y-values
Determine whether each equation defines a
function. If it is a function, give the domain.
x
Function – each x-value
corresponds to only one
y-value
a)
y = x +1
b)
y<x
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Example: Determining Whether a
Relation is a Function
Variations of the Definition of Function
Solution
1.
a) A function; since any real x input
returns a real y output, the domain is all
real x.
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.
b) Not a function.
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Function Notation
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
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
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.
Read f (2) as “f of 2” or “f at 2.”
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Example: Using Function Notation
Linear Functions
Let f (x) = x 2 + 3x – 1. Find the following.
a) f (2)
b) f (-2)
c) f (0)
d) f (2x)
Solution
A function that can be written in the form
f (x) = mx + b
for real numbers m and b is a linear function.
a) f (2) = 2 2 + 3(2) – 1 = 9
b) f (-2) = (-2)2 +3(-2) – 1 = -3
c) f (0) = 0 + 0 – 1 = –1
d) f (2x) = (2x) 2 + 3(2x) – 1 = 4x 2 + 6x – 1
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Example: Graphing Linear Functions
Graphing a Linear Function
An easy way to graph a linear function is to find 2
ordered pairs that satisfy the function.
Graph each linear function.
a) f (x) = –2x + 1
b) f (x) = 2
Solution
Plot those ordered pairs, then connect with a line.
y
y
x
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Linear Cost Models
Linear Cost Models
For linear cost function
Linear functions can be good models for a
company’s costs.
Fixed cost or up-front cost, is the cost required
before production can begin.
Variable cost is the cost to produce each item.
C(x) = mx + b
m is the variable cost,
b is the fixed cost,
x is the number of items produced.
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Example: Modeling with Linear Functions
Linear Revenue Model
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 company’s revenues can be modeled
R(x) = px
where p is the price per item,
x is the number of items sold.
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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x
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Example: Modeling with Linear Functions
Example: Modeling with Linear Functions
Solution
Solution (continued)
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.
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.
b) Each DVD sells for $20, so revenue R is given by:
R(x) = 20x.
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