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Chapter 3: Problem 32

Find a formula for the inverse function \(f^{-1}\) of the indicated function \(f\). $$ f(x)=7+8 x^{5 / 9} $$

Short Answer

Expert verified
The inverse function for \(f(x) = 7+8x^{5/9}\) is: $$ f^{-1}(x) = \left( \frac{x-7}{8} \right)^{9/5} $$

Step by step solution

01

Replace the function notation with a y-variable

Replace the function notation \(f(x)\) with a y-variable: $$ y = 7 + 8x^{5/9} $$
02

Swap x and y variables

Swap the x and y variables in the equation: $$ x = 7 + 8y^{5/9} $$
03

Solve for y in terms of x

To solve for y, we need to isolate the y-variable. Here are the substeps we will follow: 1. Subtract 7 from both sides of the equation. 2. Divide both sides by 8. 3. Raise both sides to the power of 9/5. Substep 1: Subtract 7 from both sides: $$ x - 7 = 8y^{5/9} $$ Substep 2: Divide both sides by 8: $$ \frac{x-7}{8} = y^{5/9} $$ Substep 3: Raise both sides to the power of 9/5: $$ \left( \frac{x-7}{8} \right)^{9/5} = y $$
04

Introduce the inverse function notation

Replace the y-variable with \(f^{-1}(x)\) to denote the inverse function: $$ f^{-1}(x) = \left( \frac{x-7}{8} \right)^{9/5} $$ The inverse function for \(f(x) = 7+8x^{5/9}\) is: $$ f^{-1}(x) = \left( \frac{x-7}{8} \right)^{9/5} $$

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