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Chapter 3: Problem 22

Find a formula for the inverse function \(f^{-1}\) of the indicated function \(f\). $$ f(x)=x^{12} $$

Short Answer

Expert verified
The inverse function of \(f(x) = x^{12}\) is \(f^{-1}(x) = \sqrt[12]{x}\).

Step by step solution

01

Swap \(x\) and \(y\) variables in the equation

First, replace \(f(x)\) with \(y\): $$ y = x^{12} $$ Now, swap the \(x\) and \(y\) variables to get: $$ x = y^{12} $$
02

Solve the equation for \(y\)

To find the inverse function, we need to isolate \(y\). In this case, we can do so by taking the 12th root of both sides of the equation: $$ \sqrt[12]{x} = y $$
03

Write the formula for the inverse function

Now that we have isolated \(y\), we can write the formula for the inverse function. Replace \(y\) with \(f^{-1}(x)\) to get the final formula: $$ f^{-1}(x) = \sqrt[12]{x} $$ So, the inverse function is \(f^{-1}(x) = \sqrt[12]{x}\).

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