The function $f(x)=\frac{12}{x}$ is one-to-one.
a. Find an equation for $f^{-1}(x)$, the inverse function.
b. Verify that your equation is correct by showing that $f(f^{-1}(x))=x$ and $f^{-1}(f(x))=x$.

Step 1 ：Let's find the inverse of the function $f(x)=\frac{12}{x}$. To do this, we switch the x and y (or f(x)) values and solve for y. This means we set x = $\frac{12}{y}$ and solve for y.

Step 2 ：Solving for y, we get $y=\frac{12}{x}$. This is the inverse function, $f^{-1}(x)$.

Step 3 ：We can verify this by substituting the inverse function into the original function and vice versa. If we get x as the result in both cases, then our inverse function is correct.

Step 4 ：Substituting $f^{-1}(x)$ into $f(x)$, we get $f(f^{-1}(x))=\frac{12}{f^{-1}(x)}=\frac{12}{\frac{12}{x}}=x$.

Step 5 ：Substituting $f(x)$ into $f^{-1}(x)$, we get $f^{-1}(f(x))=f^{-1}\left(\frac{12}{x}\right)=\frac{12}{\frac{12}{x}}=x$.

Step 6 ：Since $f(f^{-1}(x))=x$ and $f^{-1}(f(x))=x$, we have verified that the inverse function is correct.

Step 7 ：$\boxed{{\displaystyle f^{-1}(x)=\frac{12}{x}}}$ is the final answer.