Questions: If (f) is one-to-one and (f(14)=11), then (f^-1(11)=) and ((f(14))^-1=) .

Solution Steps

To solve this problem, we need to understand the properties of a one-to-one function and its inverse. For a one-to-one function \( f \), if \( f(a) = b \), then the inverse function \( f^{-1} \) will satisfy \( f^{-1}(b) = a \). Given \( f(14) = 11 \), we can find \( f^{-1}(11) \). The second part of the question seems to be asking for the inverse of the value \( f(14) \), which is \( 11 \).

1. Use the property of the inverse function to find \( f^{-1}(11) \).
2. Recognize that \( (f(14))^{-1} \) is simply the reciprocal of \( f(14) \).

Given that \( f(14) = 11 \), we can use the property of the inverse function. Since \( f \) is one-to-one, we have: \[ f^{-1}(11) = 14 \]

Next, we need to find the reciprocal of \( f(14) \). Since \( f(14) = 11 \), we calculate: \[ (f(14))^{-1} = \frac{1}{11} \approx 0.0909 \]

Final Answer