I know what the answer is just not how to exopress it. Can someone help me please?

State the domain of the function:

f(x)(2+x)/(3-x)

Now I know the domain is
(-infinity,3) U (3, infinity)?

(-infinity,3] U [3, infinity) ?
How do I express it? do I use ] brackets or open at the infinity and closed by the 3? i know x doesn't equal 3 so how do I show this in the answer? Thank you!

If you close at 3, then you are including 3, which will create division by zero in this case.

The variable x cannot be 3 for this rational function.

I'd say write it like this:

(-infinity,3) U (3, infinity)

The above interval notation is saying that x < 3 or x > 3.

To express the domain of a function, you have to determine all the possible values for the input variable (in this case, x) that will give you a valid output. In this case, you need to find which values of x will not result in undefined expressions.

For the given function f(x) = (2+x)/(3-x), we have to consider the denominator, (3-x). Division by zero is undefined, so we need to find any values of x that would make the denominator equal to zero. Solving the equation 3-x = 0, we get x = 3.

Since x cannot be equal to 3, we need to express that in the answer. Depending on the notation you use, you have two options:

1. Expressing the domain using open intervals: (-∞,3) ∪ (3, ∞) - This notation uses parentheses ( ) to indicate open intervals, which means the endpoint values (3 and -∞) are not included in the domain.

2. Expressing the domain using closed intervals: (-∞,3] ∪ [3, ∞) - This notation uses square brackets [ ] to indicate closed intervals, which means the endpoint values (3 and -∞) are included in the domain.

So, in your case, you can express the domain as (-∞,3) ∪ (3, ∞) or (-∞,3] ∪ [3, ∞). Both notations are correct; it depends on your preference or the specific requirements of the problem or context.