Find (f + g)x, (f-g)x, (fg)x, and (f/g)x and their domains.

f(x) = ¡î(9-x©÷), g(x) = ¡î(x©÷-4)

I cant read that Greek. Also, what in the world is (f/g) x? or (fg)x? or (f+g)x

does that last one really mean f(x)+g(x)?
Your symbols are not well understood.

To find various combinations of f(x) and g(x), we first need to evaluate each function individually.

Given:
f(x) = √(9-x²)
g(x) = √(x²-4)

To obtain the sum of f(x) and g(x), (f + g)(x), we add the two functions together.
(f + g)(x) = f(x) + g(x)
= √(9-x²) + √(x²-4)

To find the difference of f(x) and g(x), (f - g)(x), we subtract the two functions.
(f - g)(x) = f(x) - g(x)
= √(9-x²) - √(x²-4)

To determine the product of f(x) and g(x), (fg)(x), we multiply the two functions.
(fg)(x) = f(x) * g(x)
= (√(9-x²)) * (√(x²-4))
= √((9-x²) * (x²-4))

Lastly, to compute the quotient of f(x) and g(x), (f/g)(x), we divide the first function by the second.
(f/g)(x) = f(x) / g(x)
= (√(9-x²)) / (√(x²-4))

Now let's determine the domains for each of these expressions.

For (f + g)(x), (f - g)(x), and (fg)(x), we need to ensure that the square roots in the expressions are defined. Therefore, we have two conditions to consider:
1) The expression inside the square root, (9-x²) and (x²-4), should be greater than or equal to 0.
2) Denominators must not equal zero.

For (f/g)(x), we have an additional condition:
3) The expression inside the square root, (x²-4), should not equal zero.

By analyzing these conditions, we can determine the domains for each expression:

(f + g)(x): The domain is the intersection of the domains of f(x) and g(x). In this case, the domains of f(x) and g(x) are both the set of real numbers since the expressions inside the square roots are always non-negative.
Domain: All real numbers.

(f - g)(x): Same as (f + g)(x).
Domain: All real numbers.

(fg)(x): The domain is the intersection of the domains of f(x) and g(x). Again, both the expressions inside the square roots are always non-negative, so the domain remains the same.
Domain: All real numbers.

(f/g)(x): In this case, we must consider the denominator, (√(x²-4)). To avoid division by zero, we need to make sure that (x²-4) is not equal to zero.
(x²-4) ≠ 0
(x² ≠ 4)
Taking the square root of both sides where x² is positive,
|x| ≠ 2

This means the domain is all real numbers except x = ±2.
Domain: (-∞, -2) U (-2, 2) U (2, ∞)

To summarize:
(f + g)(x): Domain: All real numbers
(f - g)(x): Domain: All real numbers
(fg)(x): Domain: All real numbers
(f/g)(x): Domain: (-∞, -2) U (-2, 2) U (2, ∞)