Use function composition to determine which of the following pairs of functions are function inverses. Select all of the pairs that are function inverses.
$f(x)=\frac{1}{x}$ and $g(y)=\frac{1}{y}$
$f(x)=\frac{x-5}{3}$ and $g(y)=3(y+5)$
$f(x)=3 x+9$ and $g(y)=\frac{y-9}{3}$
$f(x)=3 x+9$ and $g(y)=\frac{y}{3}-9$
$f(x)=\frac{x-5}{3}$ and $g(y)=3 y+5$

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: \(\boxed{f(x)=\frac{1}{x} \text{ and } g(y)=\frac{1}{y}, f(x)=3 x+9 \text{ and } g(y)=\frac{y-9}{3}, f(x)=\frac{x-5}{3} \text{ and } g(y)=3 y+5}\)