Determine if functions f and g are inverses of each other: (a) \[ \begin{array}{l} f(x)=\frac{1}{4 x}, x \neq 0 \\ g(x)=\frac{1}{4 x}, x \neq 0 \\ f(g(x))=\square \\ g(f(x))=\square \end{array} \] $f$ and $g$ are inverses of each other $f$ and $g$ are not inverses of each other

Step 1 ：Given functions \(f(x) = \frac{1}{4x}\) and \(g(x) = \frac{1}{4x}\), we need to determine if they are inverses of each other.

Step 2 ：To do this, we need to check if \(f(g(x)) = x\) and \(g(f(x)) = x\).

Step 3 ：Substitute \(g(x)\) into \(f(x)\) to get \(f(g(x))\). Since \(g(x) = \frac{1}{4x}\), we have \(f(g(x)) = f(\frac{1}{4x}) = \frac{1}{4(\frac{1}{4x})} = x\).

Step 4 ：Similarly, substitute \(f(x)\) into \(g(x)\) to get \(g(f(x))\). Since \(f(x) = \frac{1}{4x}\), we have \(g(f(x)) = g(\frac{1}{4x}) = \frac{1}{4(\frac{1}{4x})} = x\).

Step 5 ：Since both \(f(g(x)) = x\) and \(g(f(x)) = x\), we can conclude that the functions \(f\) and \(g\) are inverses of each other.

Step 6 ：\(\boxed{f \text{ and } g \text{ are inverses of each other}}\)