Questions: Use algebra to find the inverse of the function f(x)=sqrt[11]x The inverse function is f^-1(x)=

Solution Steps

To find the inverse of the function \( f(x) = \sqrt[11]{x} \), we need to swap \( x \) and \( y \) and solve for \( y \). This involves raising both sides to the power of 11.

1. Start with the function \( y = \sqrt[11]{x} \).
2. Swap \( x \) and \( y \) to get \( x = \sqrt[11]{y} \).
3. Raise both sides to the power of 11 to solve for \( y \).

We start with the function defined as: \[ f(x) = \sqrt[11]{x} \]

To find the inverse function, we swap \( x \) and \( y \): \[ x = \sqrt[11]{y} \] Next, we raise both sides to the power of 11 to isolate \( y \): \[ y = x^{11} \]

Thus, the inverse function is: \[ f^{-1}(x) = x^{11} \]

To evaluate the inverse function at \( x = 2 \): \[ f^{-1}(2) = 2^{11} = 2048 \]

Final Answer