Find the domain of each function.

Question

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)

Answer 1

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).