https://file.aiquickdraw.com/mathbot/file/1726691532440m7klgz68.jpg

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.

If f is one-to-one and f(-13) = 4, then f^(-1)(4) = ? and (f(-13))^(-1) = ?

Learn how to find the inverse function f^(-1)(4) and the reciprocal of f(-13) for a one-to-one function. This step-by-step solution explores inverse functions and reciprocal concepts, ideal for students in Grades 9-12.

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