Assume that the function \( f \) is a one-to-one function. \( \begin{array}{ll}\text { 1. If } f(6)=7 \text {, find } f^{-1}(7) & \text { 2. If } f(3)=2 \text {, find } f^{-1}(2) \\ \text { 3. If } f^{-1}(-4)=-8 \text {, find } f(-8) & \text { 4. If } f^{-1}(-2)=-1 \text {, find } f(-1) \\ \text { 5. If } f(5)=2 \text {, find }(f(5))^{-1} & \text { 6. If } f(1)=4 \text {, find }(f(1))^{-1}\end{array} \)

To solve the problem, we will use the properties of one-to-one functions and their inverses. The key property we will use is that if \( f(a) = b \), then \( f^{-1}(b) = a \). Let's go through each part step by step: 1. **If \( f(6) = 7 \), find \( f^{-1}(7) \)**: - Since \( f(6) = 7 \), by the definition of the inverse function, we have \( f^{-1}(7) = 6 \). 2. **If \( f(3) = 2 \), find \( f^{-1}(2) \)**: - Since \( f(3) = 2 \), it follows that \( f^{-1}(2) = 3 \). 3. **If \( f^{-1}(-4) = -8 \), find \( f(-8) \)**: - Given \( f^{-1}(-4) = -8 \), we can deduce that \( f(-8) = -4 \). 4. **If \( f^{-1}(-2) = -1 \), find \( f(-1) \)**: - Since \( f^{-1}(-2) = -1 \), it follows that \( f(-1) = -2 \). 5. **If \( f(5) = 2 \), find \( (f(5))^{-1} \)**: - Here, \( f(5) = 2 \), so \( (f(5))^{-1} = f^{-1}(2) \). From part 2, we know \( f^{-1}(2) = 3 \). Therefore, \( (f(5))^{-1} = 3 \). 6. **If \( f(1) = 4 \), find \( (f(1))^{-1} \)**: - Since \( f(1) = 4 \), we have \( (f(1))^{-1} = f^{-1}(4) \). However, we do not have information about \( f^{-1}(4) \) from the previous parts. We cannot determine this value without additional information. In summary, the answers are: 1. \( f^{-1}(7) = 6 \) 2. \( f^{-1}(2) = 3 \) 3. \( f(-8) = -4 \) 4. \( f(-1) = -2 \) 5. \( (f(5))^{-1} = 3 \) 6. \( (f(1))^{-1} \) cannot be determined without more information.