To find the inverse function $f^{-1}(x)$ of the given function $f(x) = \left( \frac{\sqrt{x}}{7} \right)^5$, we need to follow these steps:

Step 1: Rewrite the function for simplicity

The function given is:

$f(x) = \left( \frac{\sqrt{x}}{7} \right)^5$

This can be rewritten as:

$f(x) = \frac{(\sqrt{x})^5}{7^5} = \frac{x^{5/2}}{16807}$

Step 2: Set $f(x) = y$ and solve for $x$

Let $y = f(x)$. Then:

$y = \frac{x^{5/2}}{16807}$

To solve for $x$, multiply both sides by 16807:

$16807y = x^{5/2}$

Step 3: Solve for $x$

To isolate $x$, raise both sides to the power of $\frac{2}{5}$:

$x = (16807y)^{\frac{2}{5}}$

Step 4: Express the inverse function

Since $y = f(x)$, we replace $y$ with $x$ to express the inverse function:

$f^{-1}(x) = (16807x)^{\frac{2}{5}}$

Conclusion

The inverse function is:

$\boxed{f^{-1}(x) = (16807x)^{\frac{2}{5}}}$

Would you like further details or explanations on any step?

Related Questions:

1. How do you find the inverse of a square root function?
2. What are the properties of inverse functions?
3. How does raising both sides to a fractional exponent affect the equation?
4. What is the domain and range of the function $f(x) = \left( \frac{\sqrt{x}}{7} \right)^5$?
5. How do transformations affect the graph of a function and its inverse?

Tip:

Always check if the inverse function you find is valid by substituting it back into the original function to see if you get the identity $x$.