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Chapter 2: Problem 87

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

Short Answer

Expert verified
The inverse function for the given function \(f(x) = x^9\) is \(f^{-1}(x) = \sqrt[9]{x}\).

Step by step solution

01

Represent the function in terms of y

Rewrite the given function in terms of y: \(y = x^9\)
02

Swap x and y

Switch the roles of x and y in the equation: \(x = y^9\)
03

Solve for y

Now, isolate y to find the inverse function: \(x = y^9\) Take the 9th root of both sides: \( y = \sqrt[9]{x} \)
04

Write the inverse function

Replace y with \(f^{-1}(x)\): \(f^{-1}(x) = \sqrt[9]{x}\) Thus, the inverse function for the given function is \(f^{-1}(x) = \sqrt[9]{x}\).

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