Transcript 2 6 Rational Functions I

Objectives:
1. To find the domain
of rational functions
2. To find the vertical
and horizontal
asymptotes of
rational functions
and their zeros
•
•
•
•
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Assignment:
P. 193: 5-12 (Some)
P. 193: 17-20 (Some)
P. 193: 21-26 (Some)
P. 196: 83, 84
P. 196: 85-88 (Some)
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Why is division by zero undefined?
x
1
.1
.01
.001
.0001
f(x) = 1/x
1
10
100
1000
10000
 
As x → 0+ (approaches 0 from the right), f (x)
increases without bound (approaches positive
infinity).
Why is division by zero undefined?
x
−1
−.1
−.01
−.001
−.0001
f(x) = 1/x
−1
−10
−100
−1000
−10000
 
As x → 0− (approaches 0 from the left), f (x)
decreases without bound (approaches
negative infinity).
Why is division by zero undefined?
Graphically, you can see f (x)
approach +∞ from the right
and −∞ from the left. For
this graph x = 0 is a vertical
asymptote. Also, y = 0 is a
horizontal asymptote.
Why is division by zero undefined?
This means the graph is not
continuous at x = 0. And
so the function is not
defined at x = 0. Thus,
you cannot divide by
zero.
That, plus you don’t want
to create a hole in space
and time.
Why is division by zero undefined?
Let f (x) = 2x + 5 and g(x) = 3x − 7. Find (f/g)(x)
and its domain.
A rational function can be written as
N ( x)
f ( x) 
D( x)
where N(x) and D(x) are polynomials, and
D(x) ≠ 0.
In general, the domain of a rational function is
all real numbers except the x-values that make
D(x) = 0.
N ( x)
f ( x) 
D( x)
In fact, all the interesting bits of the graphs of
rational functions occur around these
excluded x-values.
To find the domain of a rational function, follow
these 3 easy steps:
x : x  a, b
1. Set the
bottom
≠ 0.
2. Solve the
“inequation”
for x as if you
were solving a
real equation.
3. These are the
x-values that
must be
excluded
from the
domain.
Find the domain
of
each
function
below.
2
1.
9 x  9 x  10
f ( x) 
2x  8
2. f ( x) 
2x  8
9 x 2  9 x  10
Vertical Asymptote:
The line x = a is a vertical asymptote of the graph of
f (x) if f (x) →+∞ or f (x) →−∞ as x → a.
Horizontal Asymptote:
The line y = b is a horizontal asymptote of the graph of
f (x) if f (x) → b as x →+∞ or x →−∞ .
Fact: The graphs of rational functions have
asymptotes.
Query: Can the graphs of rational functions be
continuous?
Let f be the rational function given below:
N ( x) an x n  an 1 x n 1 
f ( x) 

D( x) bd x d  bd 1 x d 1 
 a1 x  a0
 b1 x  a0
Vertical Asymptotes:
1. Vertical asymptotes happen where D(x) = 0.
−
These were the values excluded from the domain! Easy!
Let f be the rational function given below:
N ( x) an x n  an 1 x n 1 
f ( x) 

D( x) bd x d  bd 1 x d 1 
 a1 x  a0
 b1 x  a0
Horizontal Asymptotes:
1. If d > n, then y = 0 is a horizontal asymptote.
–
In other words, the x-axis is a horizontal asymptote when
the degree of the denominator is greater than the
numerator.
Let f be the rational function given below:
N ( x) an x n  an 1 x n 1 
f ( x) 

D( x) bd x d  bd 1 x d 1 
 a1 x  a0
 b1 x  a0
Horizontal Asymptotes:
1. If d > n, then y = 0 is a horizontal asymptote.
–
This happens because the denominator is increasing
faster than the numerator. Thus the function approaches
zero.
Let f be the rational function given below:
N ( x) an x n  an 1 x n 1 
f ( x) 

D( x) bd x d  bd 1 x d 1 
 a1 x  a0
 b1 x  a0
Horizontal Asymptotes:
2. If d = n, then y = an/bn is a horizontal asymptote.
–
In other words, when the degrees are equal, the function
approaches the ratio of the leading coefficients.
Let f be the rational function given below:
N ( x) an x n  an 1 x n 1 
f ( x) 

D( x) bd x d  bd 1 x d 1 
 a1 x  a0
 b1 x  a0
Horizontal Asymptotes:
2. If d = n, then y = an/bn is a horizontal asymptote.
–
This happens because the numerator and denominator
are increasing/decreasing at roughly the same rate.
Let f be the rational function given below:
N ( x) an x n  an 1 x n 1 
f ( x) 

D( x) bd x d  bd 1 x d 1 
 a1 x  a0
 b1 x  a0
Horizontal Asymptotes:
3. If d < n, then there are no horizontal asymptotes.
–
This happens because the numerator is increasing faster
than the denominator.
You could always memorize the rules for finding
horizontal asymptotes, or you could just try to
understand what’s happening mathematically:
1. When d > n:
f ( x) 
f (100) 
x
x2  1
Denominator increases faster than numerator.
100
100

 .009999
10000  1 10001
Approaches 0
as x gets bigger
So y = 0 is a horizontal asymptote.
You could always memorize the rules for finding
horizontal asymptotes, or you could just try to
understand what’s happening mathematically:
2. When d = n:
2 x2
f ( x)  2
x 1
f (100) 
Both increase at roughly the same rate.
20000
20000

10000  1 10001
 1.9998
Approaches 2
as x gets bigger
So y = 2 is a horizontal asymptote.
You could always memorize the rules for finding
horizontal asymptotes, or you could just try to
understand what’s happening mathematically:
3. When d < n:
x2  1
f ( x) 
x
f (100) 
Numerator increases faster than denominator.
10000  1 10001

100
100
 100.01
Increases without
bound as x gets bigger
So there are no horizontal asymptotes.
Identify all vertical and horizontal asymptotes. Then find the
zeros of each2 function.
9 x  9 x  10
2x  8
1. f ( x) 
2. f ( x)  2
9 x  9 x  10
2x  8
Identify all vertical and horizontal asymptotes. Then find the
zeros of each function.
6 x 2  9 x  15
3x 2  21x
1. f ( x) 
2. f ( x)  2
2
3x  21x
6 x  9 x  15
Identify all vertical and horizontal asymptotes. Then find the
zeros of each function.
2 x 2  7 x  15
x2  x  6
1. f ( x)  2
2.
f ( x)  3
2 x  5 x  12
x  3x 2
Identify all vertical and horizontal asymptotes. Then find the
zeros of each function.
2 x 2  7 x  15
x2  x  6
1. f ( x)  2
2.
f ( x)  3
2 x  5 x  12
x  3x 2
A hole
Horizontal
Asymptote
Vertical
Asymptote
Horizontal
Asymptote
Vertical
Asymptote
Zero
Zero
A hole
As the previous Exercise demonstrated,
sometimes a factor in the numerator will
cancel with a factor in the denominator.
When this happens:
1. There is no vertical asymptote at the
canceled factor.
2. There is a hole in the graph at the canceled
factor, and it is not a zero.
Objectives:
1. To find the domain
of rational functions
2. To find the vertical
and horizontal
asymptotes of
rational functions
and their zeros
•
•
•
•
•
Assignment:
P. 193: 5-12 (Some)
P. 193: 17-20 (Some)
P. 193: 21-26 (Some)
P. 196: 83, 84
P. 196: 85-88 (Some)