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

Below is a step-by-step solution to each part. --- ### 1. Find \( f^{-1}(7) \) given \( f(6)=7 \). Since \( f \) is a one-to-one function, by the definition of an inverse function we have: \[ f(6)=7 \implies f^{-1}(7)=6. \] --- ### 2. Find \( f^{-1}(2) \) given \( f(3)=2 \). Using the same idea as before: \[ f(3)=2 \implies f^{-1}(2)=3. \] --- ### 3. Find \( f(-8) \) given \( f^{-1}(-4)=-8 \). The definition of the inverse function tells us: \[ f^{-1}(-4)=-8 \implies f(-8)=-4. \] --- ### 4. Find \( f(-1) \) given \( f^{-1}(-2)=-1 \). Similarly: \[ f^{-1}(-2)=-1 \implies f(-1)=-2. \] --- ### 5. Find \( (f(5))^{-1} \) given \( f(5)=2 \). Be careful: \( (f(5))^{-1} \) here denotes the reciprocal (multiplicative inverse) of the number \( f(5) \). Thus: \[ f(5)=2 \implies (f(5))^{-1}=\frac{1}{2}. \] --- ### 6. Find \( (f(1))^{-1} \) given \( f(1)=4 \). Similarly, for the reciprocal of \( f(1) \): \[ f(1)=4 \implies (f(1))^{-1}=\frac{1}{4}. \] --- ### Final Answers 1. \( f^{-1}(7)=6 \) 2. \( f^{-1}(2)=3 \) 3. \( f(-8)=-4 \) 4. \( f(-1)=-2 \) 5. \( (f(5))^{-1}=\frac{1}{2} \) 6. \( (f(1))^{-1}=\frac{1}{4} \)