The table below gives selected values for a differentiable and increasing function f and its derivative. if g(x) = f^-1(x). what is the value of g'(2) = 

x = 0, 1, 2

f(x) = 49, 2, -1

f'(x) = 0, -8, -80

To solve for \( g'(2) \) where \( g(x) = f^{-1}(x) \), we can use the inverse function derivative formula:

\[
g'(x) = \frac{1}{f'(g(x))}
\]

In this case, we need to find \( g'(2) \), which means we want the derivative of the inverse function at \( x = 2 \).

1. **Step 1: Find \( g(2) \)**  
   Since \( g(x) = f^{-1}(x) \), we are looking for an \( a \) such that \( f(a) = 2 \).
   
   From the table:
   - When \( x = 1 \), \( f(x) = 2 \).
   
   Therefore, \( g(2) = 1 \).

2. **Step 2: Calculate \( g'(2) \) using \( f'(g(2)) \)**  
   We now substitute \( g(2) = 1 \) into the derivative formula:
   
   \[
   g'(2) = \frac{1}{f'(g(2))} = \frac{1}{f'(1)}
   \]
   
   From the table, \( f'(1) = -8 \). So,

   \[
   g'(2) = \frac{1}{-8} = -\frac{1}{8}
   \]

### Final Answer:
\[
g'(2) = -\frac{1}{8}
\]

Would you like further explanation on any step? 

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**Related Questions:**

1. How do we interpret the inverse function derivative formula geometrically?
2. What happens to \( g'(x) \) when \( f'(x) = 0 \)?
3. How can we approximate \( g(x) \) values when \( f(x) \) data points are limited?
4. How would \( g'(x) \) change if \( f(x) \) were decreasing?
5. Can the inverse function theorem apply to functions that are not differentiable?

**Tip:** When finding derivatives of inverse functions, identifying points on the original function that match the given criteria simplifies calculations significantly.

The table below gives selected values for a differentiable and increasing function f and its derivative. If g(x) = f^-1(x), what is the value of g'(2)? 

x = 0, 1, 2 
f(x) = 49, 2, -1 
f'(x) = 0, -8, -80

Learn how to calculate g'(2) for the inverse of a function using given function values and derivatives. This problem covers inverse function concepts and the inverse function theorem. Suitable for advanced high school students.

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