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Find the domain of each function.

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1 Answer

Hi Paige;
1.) f(x)=(x2-4x+3)/(x-1)
At first sight, we might guess that the domain is...
(-infinity,1)U(1, +infinity)
1 must be excluded because it would render (x-1) as zero.
However, the denominator can be factored with one of the parenthetical equations as (x-1)...
f(x)=[(x-3)(x-1)]/(x-1)
 
Let's verify (x-3)(x-1) with FOIL...
FIRST...x2
OUTER...-1x
INNER...-3x
LAST...+3
x2-x-3x+3
x2-4x+3
 
f(x)=[(x-3)(x-1)]/(x-1)
Let's cancel...
f(x)=[(x-3)(x-1)]/(x-1)
f(x)=x-3
The domain is infinity.
 
2.) g(x)=(x2-4)(x-3)/(x2-x-6)
Let's factor (x2-x-6)...
Is one of the parenthetical equations (x-3)?
(x-3)(x+2)
Let's FOIL...
FIRST...x2
OUTER...2x
INNER...-3x
LAST...-6
x2+2x-3x-6
x2-x+6
(x2-4)(x-3)/(x-3)(x+2)
Let's cancel where appropriate...
(x2-4)(x-3)/(x-3)(x+2)
(x2-4)/(x+2)
Initially, it would seem that the domain is infinity excluding -2 such that the numerator can render 0.  However, it would seem that the current denominator can also be factored to include (x+2)...
(x+2)(x-2)/(x+2)
FIRST...x2
OUTER...-2x
INNER...2x
LAST...-4
Let's cancel where appropriate...
(x+2)(x-2)/(x+2)
x-2
There are no limitations on the domain.  It is from negative infinity to positive infinity.
 
3.) F(x)=(x3-2x2)/(x-2)
Can the denominator be factored to include (x-2)?
x3-2x2
x(x2-2x)
x(x-2)(x+2)
[x(x-2)(x+2)]/(x-2)
Let's cancel where appropriate...
[x(x-2)(x+2)]/(x-2)
There are no limitations on the domain.
 
4.) G(x)=√(x-1)
We already know that a negative number cannot be square rooted.  Therefore, let's begin with the one number which would set the limit...
(x-1)?0
x?1
The domain is (1, +infinity)
 
5.) f(x)=3/√(x-4)
This equation cannot be factored.
We already know that the domain cannot include 4 because that would render the numerator as 0.
We already know that a negative number cannot be square rooted.  Therefore, let's begin with the one number which would set the limit...
(x-4)?0
x?4
The domain is (4, +infinity).

Comments

Paige;
How are you doing?  I am very curious as to how you scored. Please e-mail me and let me know.
It was always my understanding that the way to establish domain is by reducing the equation to its simplest form, and then finding limitations.
I was on another website, executing some research.  According to this, the original equation is used to set the limits.
What did your instructor say?
Good Day Madam,
 
My instructor haven't said anything yet as of now. He gave us this as an assignment and it is due tomorrow. I tried to ask help from my cousin who is also good at math, and he said that my answers starting from 1 to 3 are quite different from what he expected, but rest assured that your answers from 4 to 5 are correct. :) For example, he said that the answer for number 1 is "all real numbers except 1" for number 2 he said that the domain is "all real numbers except -2 and 3" while for number 3 he said that the domain is "all real numbers except 2". But I really appreciate your hardwork for answering this! :) Please do comment if you have any ideas too :)

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