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Find the domain of each function.

Find the domain of each function.

1.) f(x)=(x^2-4x+3)/(x-1)

2.) g(x)=(x^2-4)(x-3)/(x^2-x-6)

3.) F(x)=(x^3-2x^2)/(x-2)

4.) G(x)=√(x-1)

5.) f(x)=3/√(x-4)

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).

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.