Questions: Assume that the function f is a one-to-one function. (a) if f(3)=3, find f^(-1)(3). Your answer is □ (b) If f^(-1)(-8)=-5, find f(-5). Your answer is □

Solution Steps

To solve these problems, we need to use the properties of one-to-one functions and their inverses. Specifically, for a one-to-one function \( f \) and its inverse \( f^{-1} \):

(a) If \( f(3) = 3 \), then by definition of the inverse function, \( f^{-1}(3) \) must be 3.

(b) If \( f^{-1}(-8) = -5 \), then by definition of the inverse function, \( f(-5) \) must be -8.

Given that \( f(3) = 3 \), we can use the property of inverse functions. By definition, if \( f(a) = b \), then \( f^{-1}(b) = a \). Therefore, since \( f(3) = 3 \), it follows that: \[ f^{-1}(3) = 3 \]

We are given that \( f^{-1}(-8) = -5 \). Again, using the property of inverse functions, we have: \[ f(-5) = -8 \]

Final Answer