find the domain of the composite function fog.

f(x)=x/x-2;g(x)=-8/x

a. domain of fog is {x/?}
b. the domain of fog is all real numbers

X°=1

To find the domain of the composite function f o g (fog), we need to first determine the domain of g(x) and then check any restrictions that may arise when substituting g(x) into f(x).

Let's start by determining the domain of g(x). The function g(x) is defined as g(x) = -8/x, which means that the denominator (x) cannot be equal to 0 since division by zero is undefined. Therefore, the domain of g(x) is all real numbers except x = 0.

Now, let's consider the composite function f o g (fog). The composite function is obtained by substituting g(x) into f(x). So, we have f(g(x)) = f(-8/x) = (-8/x) / ((-8/x) - 2).

To find the domain of the composite function, we need to check if there are any additional restrictions. In this case, we must ensure that the denominator of f(g(x)) is not equal to 0, as division by zero is undefined.

To find when the denominator is equal to zero, we set it equal to zero and solve for x:

(-8/x) - 2 = 0
-8 - 2x = 0
-2x = 8
x = -4

Therefore, x = -4 is an additional restriction for the composite function f o g (fog).

To summarize:
a. The domain of fog is {x ≠ 0, x ≠ -4}, which means all real numbers except x = 0 and x = -4.
b. The statement "the domain of fog is all real numbers" is incorrect because the additional restriction x = -4 must be considered.