Question

if f is a one - to - one function such that f(5)=-3 and f(5)=7, what is (f^(-1))(-3)? enter your answer as a reduced fraction.

Explanation:

Step1: Recall inverse - function derivative formula

The formula for the derivative of the inverse function is $(f^{-1})'(y)=\frac{1}{f'(x)}$, where $y = f(x)$.

Step2: Identify $x$ and $y$ values

We are given that $f(5)=-3$ and $f'(5) = 7$. We want to find $(f^{-1})'(-3)$. Here, $x = 5$ and $y=-3$ since $y = f(x)$ and we want to find the derivative of the inverse function at $y=-3$.

Step3: Apply the formula

Using the formula $(f^{-1})'(-3)=\frac{1}{f'(5)}$.

Step4: Substitute the given value

Since $f'(5)=7$, then $(f^{-1})'(-3)=\frac{1}{7}$.

Answer:

$\frac{1}{7}$