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Chapter 3: Problem 26

Find a formula for the inverse function \(f^{-1}\) of the indicated function \(f\). $$ f(x)=x^{-17 / 7} $$

Short Answer

Expert verified
The inverse function of \(f(x) = x^{-\frac{17}{7}}\) is \(f^{-1}(x) = x^{-\frac{7}{17}}\).

Step by step solution

01

Replace f(x) with y

Let's replace the function notation with the variable y: $$ y = x^{-17 / 7} $$
02

Swap x and y positions

Now, we'll switch the positions of x and y: $$ x = y^{-17 / 7} $$
03

Solve for y

In this step, we'll isolate the y variable by raising both sides of the equation by -7/17, which is the reciprocal of -17/7: $$ x^{-\frac{7}{17}} = y^{\frac{-17}{7} \cdot \frac{-7}{17}} = y $$
04

Write the inverse function

Finally, we can write the equation for the inverse function, \(f^{-1}(x)\): $$ f^{-1}(x) = x^{-\frac{7}{17}} $$ Therefore, the inverse function of \(f(x) = x^{-\frac{17}{7}}\) is \(f^{-1}(x) = x^{-\frac{7}{17}}\).

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