Let's analyze the given problem:

1. We are told that $f$ is a one-to-one function and $f(-13) = 4$.

2. The first part asks for $f^{-1}(4)$. The inverse function, $f^{-1}$, reverses the operation of $f$. This means that $f^{-1}(4)$ returns the value $x$ such that $f(x) = 4$. From the problem, we know $f(-13) = 4$, so $f^{-1}(4) = -13$.

   Answer: $f^{-1}(4) = -13$

3. The second part asks for $\left( f(-13) \right)^{-1}$. Since $f(-13) = 4$, this expression becomes $4^{-1}$. The inverse of 4, as a number, is $\frac{1}{4}$.

   Answer: $\left( f(-13) \right)^{-1} = \frac{1}{4}$.

Let me know if you want further explanation.

Related Questions:

1. What does it mean for a function to be one-to-one?
2. How do you determine the inverse of a function in general?
3. Why is the inverse of a number simply its reciprocal?
4. Can you always find the inverse of a function?
5. What is the difference between the inverse of a function and the reciprocal of a number?

Tip: For one-to-one functions, each output corresponds to exactly one input, making finding inverses straightforward.