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Chapter 3

Functions and Graphs
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Section 3.1

Functions
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Example:
Solution:
Solution:
Solution:
Each of the given equations defines y as a function of x. Find
the domain of each function.
(a) y  x 4
Any number can be raised to the fourth power, so the domain is the
set of all real numbers, which is sometimes written as  ,   .
(b) y  x  6
For y to be a real number, x  6 must be nonnegative. This happens
only when x  6  0 —that is, when x  6. So the domain is the
interval  6,   .
(c) y 
1
x3
Because the denominator cannot be 0, x  3 and the domain
consists of all numbers in the intervals,
 ,  3
or
 3,   .
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Section 3.2

Graphs and Functions
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Example:
Solution:
Graph the absolute-value function, whose rule is f ( x )  x .
The absolute value function can
be defined as the piecewise
function
if x  0
x
f ( x)  
  x if x  0.
 0,0 
So the right half of the graph (that is, where x  0) will consist of a
portion of the line y  x . It can be graphed by plotting two points, say
 0, 0  and 1, 1.
The left half of the graph (where x  0) will consist of a portion of
the line y   x, which can be graphed by plotting  2, 2  and  1, 1 .
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Section 3.3

Applications of Linear Functions
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Example:
Solution:
An anticlot drug can be made for $10 per unit. The total cost to
produce 100 units is $1500.
(a) Assuming that the cost function is linear, find its rule.
Since the cost function is linear, its rule is of the form C ( x)  mx  b.
We are given that m (the cost per item) is 10, so the rule is C ( x)  10 x  b.
To find b, use the fact that it costs $1500 to produce 100 units
which means that
C (100)  1500
10(100)  b  1500
Solution:
1000  b  1500
b  500.
So the rule is C ( x)  10 x  500.
(b) What are the fixed costs?
The fixed costs are C (0)  10(0)  500  $500.
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Section 3.4

Quadratic Functions
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Example:
Solution:
Graph each of these quadratic functions:
f ( x)  x 2 ;
g ( x)  4 x 2 ;
h( x)  .2 x 2 .
In each case, choose several numbers (negative, positive, and 0) for
x, find the values of the function at these numbers, and plot the
corresponding points. Then connect the points with a smooth curve
to obtain the graphs.
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Section 3.5

Polynomial Functions
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Example:
Solution:
Graph f ( x)  x 3 .
First, find several ordered pairs belonging to the graph. Be sure to
choose some negative x-values, x  0, and some positive x-values
in order to get representative ordered pairs. Find as many ordered
pairs as you need in order to see the shape of the graph and draw a
smooth curve through them to obtain the graph below.
f ( x)  x3
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Section 3.6

Rational Functions
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Example:
Solution:
Graph
2 x2
f ( x)  2
.
x 4
Find the vertical asymptotes by setting the denominator equal to 0
and solving for x:
x2  4  0
 x  2  x  2   0
x  2  0 or
x  2 or
x20
x  2.
Factor.
Set each term equal to 0.
Solve for x.
Since neither of these numbers makes the numerator 0, the lines
x  2 and x  2 are vertical asymptotes of the graph.
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Example:
Solution:
Graph
2 x2
f ( x)  2
.
x 4
The horizontal asymptote can be
determined by dividing both the numerator
and denominator of f ( x) by x 2(the highest
power of x that appears in either one).
2 x2
f ( x)  2
x 4
2 x2
2
x
f ( x)  2
x 4
x2
2x2
2
x
f ( x)  2
x
4

x2 x2
2
f ( x) 
4
1 2
x
When x is very large, the
fraction 4 / x 2 is very close to 0,
so the denominator is very close
to 1 and f ( x) is very close to 2.
Hence, the line y  2 is the
horizontal asymptote of the graph.
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Example:
Graph
2 x2
f ( x)  2
.
x 4
Solution: Using this information and plotting several points in each of the three
regions defined by the vertical asymptotes, we obtain the desired graph.
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