For the function f, of, x, equals, left parenthesis, start fraction, square root of, x, end square root, divided by, 7, end fraction, right parenthesis, to the power 5f(x)=(
7
x
​

​
)
5
, find f, to the power minus 1 , left parenthesis, x, right parenthesisf
−1
(x).

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 \( x \).

For the function f(x) = (sqrt(x) / 7)^5, find the inverse function f^(-1)(x).

Learn how to find the inverse function of f(x) = (sqrt(x) / 7)^5, with detailed step-by-step explanations. This solution covers key concepts in algebra, exponents, and radicals, making it suitable for students in Grades 11-12 or early college.

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