Question

1. given the following function $f(x) = \frac{1}{2}x - 4$

a) find $f^{-1}(x)$

b) find $f(f^{-1}(-3))$.

Explanation:

Step1: Recall the property of inverse functions

For a function \( f(x) \) and its inverse \( f^{-1}(x) \), the composition \( f(f^{-1}(x)) = x \) for all \( x \) in the domain of \( f^{-1} \) and \( f^{-1}(f(x)) = x \) for all \( x \) in the domain of \( f \). This is because the inverse function undoes the operation of the original function.

Step2: Apply the property to the given problem

We need to find \( f(f^{-1}(-3)) \). Using the property \( f(f^{-1}(x)) = x \), we can substitute \( x = -3 \). So \( f(f^{-1}(-3)) = -3 \).

Answer:

\(-3\)