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Chapter 3: Problem 24

Find a formula for the inverse function \(f^{-1}\) of the indicated function \(f\). $$ f(x)=x^{1 / 11} $$

Short Answer

Expert verified
The inverse function of the given function \(f(x) = x^{1/11}\) is \(f^{-1}(x) = x^{11}\).

Step by step solution

01

Replace \(f(x)\) with \(y\)

We replace \(f(x)\) with \(y\) to rewrite the function in terms of \(y\): $$ y = x^{1/11} $$
02

Swap \(x\) and \(y\)

Swapping \(x\) and \(y\) in the equation, as we want to find the inverse function, we get: $$ x = y^{1/11} $$
03

Solve for \(y\)

To solve for \(y\), we will raise both sides of the equation to the power of 11, as we want to eliminate the exponent \(1/11\) in the equation. So, $$ x^{11} = (y^{1/11})^{11} $$ Simplifying the right side, we obtain: $$ x^{11} = y $$
04

Write the final result

The final result should be written as a function of \(x\). The inverse function of \(f(x)\), denoted by \(f^{-1}(x)\), is: $$ f^{-1}(x) = x^{11} $$

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