Question

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:

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