Question: Assume that $f(x)$ is a one-to-one differentiable function with values given on the following table.
$x$ -3 1 2 3
$f(x)$ 3 -3 1 2
$f'(x)$ -1 8 3 2
Find the following values.
(a) $f^{-1}(3) =$
(b) $(f^{-1})'(3) =$
1. **State the problem:** We are given a one-to-one differentiable function $f(x)$ with values and derivatives at certain points. We need to find:
(a) $f^{-1}(3)$, the value of the inverse function at 3.
(b) $(f^{-1})'(3)$, the derivative of the inverse function at 3.
2. **Recall the inverse function definition:**
If $y = f(x)$, then $x = f^{-1}(y)$.
So, $f^{-1}(3)$ is the $x$ such that $f(x) = 3$.
3. **Find $f^{-1}(3)$:**
From the table, $f(-3) = 3$.
Therefore, $f^{-1}(3) = -3$.
4. **Formula for the derivative of the inverse function:**
$$(f^{-1})'(y) = \frac{1}{f'(f^{-1}(y))}$$
This means to find $(f^{-1})'(3)$, we need $f'(x)$ where $x = f^{-1}(3)$.
5. **Calculate $(f^{-1})'(3)$:**
We found $f^{-1}(3) = -3$.
From the table, $f'(-3) = -1$.
So,
$$ (f^{-1})'(3) = \frac{1}{f'(-3)} = \frac{1}{-1} = -1 $$
**Final answers:**
(a) $f^{-1}(3) = -3$
(b) $(f^{-1})'(3) = -1$