find the domain of the composite function fog.

f(x)= 2/x-3; g(x)= 7/x

please show all work.

To find the domain of the composite function fog, we need to consider the restrictions of both the functions f(x) and g(x) individually, as well as the restrictions imposed by the composition.

Given:
f(x) = 2/(x-3)
g(x) = 7/x

1. Determine the domain of f(x):
The function f(x) has a restriction when the denominator of a fraction becomes zero since division by zero is undefined.
In this case, the denominator x-3 must not equal zero, so we solve for x:
x - 3 ≠ 0
x ≠ 3

Therefore, the domain of f(x) is all real numbers except x = 3.

2. Determine the domain of g(x):
The function g(x) has a restriction when the denominator becomes zero.
In this case, the denominator x must not equal zero, so we solve for x:
x ≠ 0

Therefore, the domain of g(x) is all real numbers except x = 0.

3. Determine the domain of fog:
To find the domain of the composite function fog, we need to consider the composition f(g(x)).

First, we substitute g(x) into f(x). So instead of x in f(x), we put g(x):
f(g(x)) = 2/(g(x) - 3)

Now we replace g(x) with its expression:
f(g(x)) = 2/(7/x - 3)

To simplify, we get a common denominator:
f(g(x)) = 2/((7 - 3x)/x)

We can invert the division and multiply by the reciprocal:
f(g(x)) = 2*(x/(7 - 3x))

Now, we need to consider any additional restrictions. The expression 7 - 3x cannot be equal to zero as it would make the denominator zero and result in an undefined value. So we solve for x:
7 - 3x ≠ 0
-3x ≠ -7
x ≠ 7/3

Therefore, the domain of the composite function fog is all real numbers except x = 3 and x = 7/3.