Final answer:
The given functions f(x) = 1/(x - 5) and g(x) = 5x + 1/x are not inverses of each other as the compositions f(g(x)) and g(f(x)) do not yield the identity function for all x. The domain for the composition of these functions is (-infinity, 5) U (5, infinity).
Step-by-step explanation:
To determine if the given functions f(x) = 1/(x - 5) and g(x) = 5x + 1/x are inverses of each other, we need to use function composition. We will find the compositions f(g(x)) and g(f(x)) and see if they yield the identity function, which is f(g(x)) = x and g(f(x)) = x for all x in the domains of the respective functions.
Let's compose f(g(x)):
f(g(x)) = f(5x + 1/x) = 1 / ((5x + 1/x) - 5) = 1 / (5x - 5 + 1/x) = 1 / (5(x - 1) + 1/x).
This does not simplify to x, meaning f and g are not inverses.
Now, let's check g(f(x)):
g(f(x)) = g(1/(x - 5)) = 5(1/(x - 5)) + 1/(1/(x - 5)) = 5/(x - 5) + (x - 5)/1 = (5 + x - 5)/(x - 5) = x/(x - 5).
Again, this does not simplify to x, confirming that f and g are not inverses. The domain of both f(x) and g(x) excludes the point where x = 5 since it would result in a division by zero.
Based on the function composition it is clear that f(x) and g(x) are not inverses of each other, and the domain for the composition of these functions, where both are defined, is (-infinity, 5) U (5, infinity).