For the function f(x)=7(x)^1/7, find the f^-1(x)

Answer 1

To find the inverse function \( f^{-1}(x) \) of the function \( f(x) = 7x^{1/7} \), we will follow these steps:

1. Replace \( f(x) \) with \( y \):

   \[ y = 7x^{1/7} \]

2. Swap \( x \) and \( y \):

   \[ x = 7y^{1/7} \]

3. Solve for \( y \):

   First, isolate \( y^{1/7} \):

   \[ y^{1/7} = \frac{x}{7} \]

   Now raise both sides to the power of 7:

   \[ y = \left(\frac{x}{7}\right)^7 \]

4. Write the inverse function:

   \[ f^{-1}(x) = \left(\frac{x}{7}\right)^7 \]

Thus, the inverse function is:

\[ f^{-1}(x) = \frac{x^7}{7^7} \]