Step 1 :Let's start with the first pair of functions: \(f(x)=\frac{x}{3}\) and \(g(x)=3x\).
Step 2 :Find the composition \(f(g(x))\), which simplifies to \(\frac{3x}{3} = x\).
Step 3 :Find the composition \(g(f(x))\), which simplifies to \(3\left(\frac{x}{3}\right) = x\).
Step 4 :Since both \(f(g(x))\) and \(g(f(x))\) simplify to \(x\), \(f\) and \(g\) are inverses of each other.
Step 5 :Next, let's move on to the second pair of functions: \(f(x)=2x-3\) and \(g(x)=\frac{x+3}{2}\).
Step 6 :Find the composition \(f(g(x))\), which simplifies to \(2\left(\frac{x+3}{2}\right) - 3 = x\).
Step 7 :Find the composition \(g(f(x))\), which simplifies to \(\frac{2x-3+3}{2} = x\).
Step 8 :Since both \(f(g(x))\) and \(g(f(x))\) simplify to \(x\), \(f\) and \(g\) are inverses of each other.
Step 9 :Final Answer: \(\boxed{\text{For the first pair of functions, } f(x)=\frac{x}{3} \text{ and } g(x)=3x, f \text{ and } g \text{ are inverses of each other.}}\)
Step 10 :Final Answer: \(\boxed{\text{For the second pair of functions, } f(x)=2x-3 \text{ and } g(x)=\frac{x+3}{2}, f \text{ and } g \text{ are inverses of each other.}}\)