Final answer:
The functions f(x) = 5/(x - 9) and g(x) = 9/x are not inverses of each other, as their composition does not result in the identity function.
Step-by-step explanation:
The question asks whether the given functions f(x) = 5/(x - 9) and g(x) = 9/x are inverses of each other. To determine if two functions are inverses, one must check if composing them results in the identity function, meaning f(g(x)) = x and g(f(x)) = x. Let's perform the composition of these functions.
First, calculate f(g(x)):
f(g(x)) = f(9/x) = 5/((9/x) - 9) = 5/(9/x - 9x/x) = 5/(9/x - 9/1) = 5/(9 - 9x)/x.
As we can see, f(g(x)) does not simplify to x.
Now, calculate g(f(x)):
g(f(x)) = g(5/(x - 9)) = 9/(5/(x - 9)) = 9*(x - 9)/5 = (9x - 81)/5.
Similarly, g(f(x)) does not simplify to x either.
Therefore, the functions f(x) and g(x) are not inverses of each other since the composition of the functions does not result in the identity function.