For each pair of functions $f$ and $g$ below, find $f(g(x))$ and $g(f(x))$. Then, determine whether $f$ and $g$ are inverses of each other. Simplify your answers as much as possible. (Assume that your expressions are defined for all $x$ in the domain of the composition. You do not have to indicate the domain.) (a) \[ \begin{array}{l|l} f(x)=\frac{4}{x}, x \neq 0 & \text { (b) } f(x)=6 x+1 \\ g(x)=\frac{4}{x}, x \neq 0 & g(x)=6 x-1 \\ f(g(x))=\square & f(g(x))=\square \\ g(f(x))=\square & g(f(x))=\square \end{array} \] $f$ and $g$ are inverses of each other $f$ and $g$ are not inverses of each other $f$ and $g$ are inverses of each other $f$ and $g$ are not inverses of each other

Step 1 ：We are given two pairs of functions and we are asked to find the composition of these functions and determine if they are inverses of each other.

Step 2 ：For the first pair, we have \(f(x) = \frac{4}{x}\) and \(g(x) = \frac{4}{x}\). To find \(f(g(x))\) and \(g(f(x))\), we need to substitute \(g(x)\) into \(f(x)\) and vice versa.

Step 3 ：For the second pair, we have \(f(x) = 6x + 1\) and \(g(x) = 6x - 1\). Similarly, we need to substitute \(g(x)\) into \(f(x)\) and vice versa.

Step 4 ：Two functions are inverses of each other if and only if \(f(g(x)) = x\) and \(g(f(x)) = x\) for all \(x\) in their domains.

Step 5 ：Let's start with the first pair of functions. After substitution, we find that \(f(g(x)) = x\) and \(g(f(x)) = x\). This means that \(f\) and \(g\) are inverses of each other for the first pair.

Step 6 ：Now, let's move on to the second pair of functions. After substitution, we find that \(f(g(x)) = 36x - 5\) and \(g(f(x)) = 36x + 5\). This means that \(f\) and \(g\) are not inverses of each other for the second pair because \(f(g(x)) \neq x\) and \(g(f(x)) \neq x\).

Step 7 ：\(\boxed{\text{Final Answer: For the first pair of functions, } f \text{ and } g \text{ are inverses of each other. For the second pair of functions, } f \text{ and } g \text{ are not inverses of each other.}}\)