Final answer:
The domain of the composition (f o g)(x) is all real numbers except x = 7, because within the composition, g(x) excludes this value from the domain and no additional values are excluded when g(x) is composed with f(x).
Step-by-step explanation:
To find the domain of the composition (f ∘ g)(x), we must consider the domains of f(x) and g(x) individually and how they interact when combined. For f(x) = 1/(7x-6), the domain excludes values of x that make the denominator zero, in this case, x = 6/7. Similarly, g(x) = x/(x-7) has a domain that excludes x = 7, since that would also make the denominator zero.
When composing (f ∘ g)(x), first g(x) is applied, which prohibits x = 7 from the domain. Then the result of g(x) is used as the input to f(x). Substituting g(x) into f(x) gives f(g(x)) = 1/(7(x/(x-7)) - 6). To find where this function is undefined, we set the denominator equal to zero and solve for x: 7(x/(x-7)) - 6 = 0. Simplifying, we reach a point where x = 6 as an excluded value, but this is impossible because x = 7 is already excluded by g(x)'s domain.
Therefore, the domain of the composition (f ∘ g)(x) is all real numbers except x = 7. Option A is the correct answer.