Step 1 :Two functions are inverses of each other if and only if the composition of the two functions in both orders results in the identity function. The identity function is a function that always returns the same value that was used as its argument. In other words, for all x, Identity(x) = x. So, we need to check if \(f(g(x)) = x\) and \(g(f(x)) = x\) for all x in the domain of the functions.
Step 2 :Let's check the first pair of functions: \(f(x)=\frac{1}{x}\) and \(g(y)=\frac{1}{y}\).
Step 3 :Let's check the second pair of functions: \(f(x)=\frac{x-5}{3}\) and \(g(y)=3(y+5)\).
Step 4 :Let's check the third pair of functions: \(f(x)=3 x+9\) and \(g(y)=\frac{y-9}{3}\).
Step 5 :Let's check the fourth pair of functions: \(f(x)=3 x+9\) and \(g(y)=\frac{y}{3}-9\).
Step 6 :Let's check the fifth pair of functions: \(f(x)=\frac{x-5}{3}\) and \(g(y)=3 y+5\).
Step 7 :After checking all the pairs, we find that the pairs of functions that are inverses of each other are \(f(x)=\frac{1}{x}\) and \(g(y)=\frac{1}{y}\), \(f(x)=3 x+9\) and \(g(y)=\frac{y-9}{3}\), and \(f(x)=\frac{x-5}{3}\) and \(g(y)=3 y+5\).
Step 8 :Final Answer: \(\boxed{f(x)=\frac{1}{x} \text{ and } g(y)=\frac{1}{y}, f(x)=3 x+9 \text{ and } g(y)=\frac{y-9}{3}, f(x)=\frac{x-5}{3} \text{ and } g(y)=3 y+5}\)