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Chapter 3: Problem 23

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

Short Answer

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

Step by step solution

01

Replace function notation with a variable

Replace the function notation \(f(x)\) with a variable, in this case, \(y\). So, our equation becomes: $$ y = x^{\frac{1}{7}} $$
02

Switch the roles of x and y

Now, we will switch the roles of \(x\) and \(y\). This means we will replace \(y\) with \(x\) and \(x\) with \(y\). So, the equation becomes: $$ x = y^{\frac{1}{7}} $$
03

Solve for y

Next, we will solve the equation for \(y\). We can do this by taking both sides of the equation to the power of 7: $$ (x)^{7} = (y^{\frac{1}{7}})^{7} $$ Simplifying the equation, we obtain: $$ x^7 = y $$
04

Replace y with inverse function notation

Finally, replace the variable \(y\) with the inverse function notation \(f^{-1}(x)\), so the inverse function is: $$ f^{-1}(x) = x^{7} $$ So, the inverse function of \(f(x) = x^{\frac{1}{7}}\) is \(f^{-1}(x) = x^{7}\).

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