Question

MAX POINTS TO BEST ANSWER!

Find the domain of the function (f/g)(x) where f(x)=x^2-9 and g(x)=x^2-4x+3

Please help me understand how to find the answer, I do know the answer as (-infinity,1) union (1,3) union (3,infinity) however, I do not know how to get to the answer so if someone could explain it to me tat would be amazing! Thanks :)

Answer 1

Note: (f/g)(x) can be written as

f(x)/g(x). This simply means f(x) divided by g(x).

We know what f(x) and g(x) are because that information is given to us in the problem. See it?

Set up the rational function and factor.

(x^2 - 9)/(x^2 -4x + 3)

We now factor the top and bottom.

(x - 3)(x + 3)/(x - 3)(x - 1)

Cancel out common terms.

We now have (x + 3)/(x - 1) as the reduced rational function.

To find the domain, set the bottom to 0 and solve for x.

x - 1 = 0

x = 1

The domain is ALL REAL NUMBERS EXCEPT FOR 1. In the given function, x CANNOT BE 3 because this would create a zero in the denominator.

Answer in interval notation is

(-00, 1) U (1, 3) U (1, 00).

The interval notation reads as follows:

All negative numbers included but not 1 united with the open interval that does not include 1 and 3 united with the interval that excludes 1 but includes all positive numbers greater than 1. I hope this is clear.