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We are given a table for a differentiable and increasing function \( f(x) \) and its derivative \( f'(x) \). We are asked to find \( g'(0) \), where \( g(x) = f^{-1}(x) \) is the inverse function of \( f(x) \). 

### Key Information:
1. \( g(x) = f^{-1}(x) \)
2. The derivative of the inverse function can be expressed as:
   \[
   g'(x) = \frac{1}{f'(g(x))}
   \]
3. We are asked to find \( g'(0) \), meaning we need to find \( g'(0) \).

### Steps:
1. Since \( g(x) = f^{-1}(x) \), we know \( f(g(x)) = x \).
2. To find \( g'(0) \), first locate the value of \( f \) such that \( f(g(0)) = 0 \). From the table, \( f(0) = 3 \), so \( g(3) = 0 \), meaning \( g(0) = -4 \).
3. Now, use the formula for the derivative of the inverse:
   \[
   g'(0) = \frac{1}{f'(g(0))} = \frac{1}{f'(-4)}
   \]
4. From the table, \( f'(-4) = 1 \), so:
   \[
   g'(0) = \frac{1}{1} = 1
   \]

Thus, the correct answer is option **C**.

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Would you like any further clarifications or details?

Here are 5 related questions to expand your understanding:
1. How do you compute the derivative of an inverse function in general?
2. What are the conditions for a function to have an inverse?
3. How does the relationship between a function and its inverse affect their derivatives?
4. Can this method of finding the derivative of an inverse be applied to non-differentiable functions?
5. What would change in the calculation if the function \( f(x) \) were not increasing?

**Tip:** When working with inverse functions, always check how the function and its inverse are related through specific values, especially at critical points.

The table above gives selected values for a differentiable and increasing function f and its derivative f'. If g(x) = f⁻¹(x) for all x, which of the following is a correct expression for g'(0)?

Learn how to compute the derivative of the inverse function g(x) = f⁻¹(x) using the derivative rule for inverses. This example focuses on finding g'(0) with the provided table of values for f(x) and f'(x). Suitable for high school students (Grades 11-12) studying calculus.

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