Theresa L.

asked • 10/12/13

Find the domain of the rational function please

Find the domain of the rational expression (Show all work)
 
R(x)= -7-6x/x^3-5x^2+6x

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Michael F. answered • 10/12/13

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R(x)= -7-6x/x^3-5x^2+6x submitted, I am going to assume you meant
R(x)= (-7-6x)/(x³-5x²+6x)=-(7+6x)/(x(x²-5x+6)=-(7+6x)/(x(x-3)(x-2))
The function is definable except where x=0 or 2 or 3.
The domain D is the complement, in the real line, of these points.
D=(-∞,0)∪(0,2)υ(2,3)∪(3,∞)
 
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Vivian L. answered • 10/12/13

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Hi Again Theresa;

I am assuming this...

R(x)= -7-6x/x3-5x2+6x

Is this...

R(x)= (-7-6x)/(x3-5x2+6x)

We initially know that...

x cannot equal 0.  Henceforth, this is not part of the domain.

Let's simplify the numerator...

(x3-5x2+6x)

x(x2-5x+6)

 

(x2-5x+6) cannot equal zero.

Let's factor...

For the LAST of FOIL, 6 x 1, or 3 x 2

The OUTER and INNER of FOIL must add-up to be -5x.

Henceforth, we will use -3 and -2...

(x-3)(x-2)

 

x cannot equal 3

x cannot equal 2

x cannot equal 0

 

Let's check our results with the original numerator...

x3-5x2+6x

x=3

33-5(32)+6(3)=27-5(9)+18=45-45=0

x=2

23-5(22)+6(2)=8-5(4)+12=20-20=0

The domain is...

(-infinity, 0)(1)(4, +infinity)

 

 
 
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Michael F.

Your assessment of the domain as
x cannot equal 3 and x cannot equal 2 and x cannot equal 0
is clearly correct, however your description should be the domain is (-∞,0)∪(0,2)∪(2,3)∪(3,∞), the parentheses indicating that the intervals do not contain their end points
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10/14/13

Vivian L.

Thank you!
If I understand correctly, the U denotes the fact that "the parentheses indicate that the intervals do not contain their end points".
If there is no U, then the parentheses indicate that the intervals are inclusive of their end points.
I never before encountered the symbol of U.
Report

10/15/13

Vivian L.

Thank you!
If I understand correctly, the U denotes the fact that "the parentheses indicate that the intervals do not contain their end points".
If there is no U, then the parentheses indicate that the intervals are inclusive of their end points.
I never before encountered the symbol of U.
Report

10/15/13

Vivian L.

Thank you!
If I understand correctly, the U denotes the fact that "the parentheses indicate that the intervals do not contain their end points".
If there is no U, then the parentheses indicate that the intervals are inclusive of their end points.
I never before encountered the symbol of U.
Report

10/15/13

Vivian L.

Thank you!
If I understand correctly, the U denotes the fact that "the parentheses indicate that the intervals do not contain their end points".
If there is no U, then the parentheses indicate that the intervals are inclusive of their end points.
I never before encountered the symbol of U.
Report

10/15/13

Vivian L.

Thank you!
If I understand correctly, the U denotes the fact that "the parentheses indicate that the intervals do not contain their end points".
If there is no U, then the parentheses indicate that the intervals are inclusive of their end points.
I never before encountered the symbol of U.
Report

10/15/13

Vivian L.

Thank you!
If I understand correctly, the U denotes the fact that "the parentheses indicate that the intervals do not contain their end points".
If there is no U, then the parentheses indicate that the intervals are inclusive of their end points.
I never before encountered the symbol of U.
Report

10/15/13

Vivian L.

Thank you!
If I understand correctly, the U denotes the fact that "the parentheses indicate that the intervals do not contain their end points".
If there is no U, then the parentheses indicate that the intervals are inclusive of their end points.
I never before encountered the symbol of U.
Report

10/15/13

Vivian L.

Thank you!
If I understand correctly, the U denotes the fact that "the parentheses indicate that the intervals do not contain their end points".
If there is no U, then the parentheses indicate that the intervals are inclusive of their end points.
I never before encountered the symbol of U.
Report

10/15/13

Vivian L.

Thank you!
If I understand correctly, the U denotes the fact that "the parentheses indicate that the intervals do not contain their end points".
If there is no U, then the parentheses indicate that the intervals are inclusive of their end points.
I never before encountered the symbol of U.
Report

10/15/13

Vivian L.

Thank you!
If I understand correctly, the U denotes the fact that "the parentheses indicate that the intervals do not contain their end points".
If there is no U, then the parentheses indicate that the intervals are inclusive of their end points.
I never before encountered the symbol of U.
Report

10/15/13

Vivian L.

Thank you!
If I understand correctly, the U denotes the fact that "the parentheses indicate that the intervals do not contain their end points".
If there is no U, then the parentheses indicate that the intervals are inclusive of their end points.
I never before encountered the symbol of U.
Report

10/15/13

Michael F.

No! the parentheses ( and ) indicate no endpoints.  The symbol ∪ means set union. 
Michael F.
Report

10/15/13

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