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Chapter 3: Problem 31

Find a formula for the inverse function \(f^{-1}\) of the indicated function \(f\). $$ f(x)=4 x^{3 / 7}-1 $$

Short Answer

Expert verified
The inverse function of \(f(x) = 4x^{3/7} - 1\) is \(f^{-1}(x) = \left(\frac{x + 1}{4}\right)^{7/3}\).

Step by step solution

01

Rewrite the function using y

Rewrite the given function with \(y\) instead of \(f(x)\): $$ y = 4x^{3/7} - 1 $$
02

Swap x and y variables

Swap the \(x\) and \(y\) variables in the equation: $$ x = 4y^{3/7} - 1 $$
03

Solve for y in terms of x

To solve for \(y\), follow these steps: 1. Add \(1\) to both sides of the equation: $$ x + 1 = 4y^{3/7} $$ 2. Divide both sides by \(4\): $$ \frac{x + 1}{4} = y^{3/7} $$ 3. Raise both sides to the power of \(7/3\) to eliminate the exponent on \(y\): $$ \left(\frac{x + 1}{4}\right)^{7/3} = y $$ Now, we have found the inverse function: $$ f^{-1}(x) = \left(\frac{x + 1}{4}\right)^{7/3} $$

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