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Chapter 3: Problem 27

Find a formula for the inverse function \(f^{-1}\) of the indicated function \(f\). $$ f(x)=\frac{x^{4}}{81} $$

Short Answer

Expert verified
The short answer for finding the inverse function \(f^{-1}(x)\) of the given function \(f(x)=\frac{x^4}{81}\) is: 1. Replace \(f(x)\) with \(y\): \(y = \frac{x^4}{81}\) 2. Swap the roles of \(x\) and \(y\): \(x = \frac{y^4}{81}\) 3. Solve for \(y\): \(y = \sqrt[4]{81x}\) 4. Write the final inverse function: \(f^{-1}(x) = \sqrt[4]{81x}\)

Step by step solution

01

Rewrite the function

Replace \(f(x)\) with \(y\): $$ y = \frac{x^4}{81} $$
02

Switch x and y

Swap the roles of \(x\) and \(y\): $$ x = \frac{y^4}{81} $$
03

Solve for y

We need to isolate \(y\) by performing the following operations: 1. Multiply both sides by 81 to get rid of the denominator: $$ 81x = y^4 $$ 2. Take the fourth root of both sides to isolate \(y\): $$ y = \sqrt[4]{81x} $$
04

Write the inverse function

Replace \(y\) with \(f^{-1}(x)\) and write the final inverse function: $$ f^{-1}(x) = \sqrt[4]{81x} $$

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