The inverse of the function `f (x) = [1 -(x - 5)^3 ]^(1//5)` is
`5+ (1-x^5)^(1//3)`

Answer

To find the inverse of the function \( f(x) = [1 - (x - 5)^3]^{1/5} \), we will follow these steps:

### Step 1: Set \( f(x) \) equal to \( y \)
Let \( y = f(x) \).
\[ y = [1 - (x - 5)^3]^{1/5} \]

### Step 2: Eliminate the fifth root
To eliminate the fifth root, raise both sides to the power of 5.
\[ y^5 = 1 - (x - 5)^3 \]

### Step 3: Rearrange the equation
Now, rearranging the equation to isolate the term involving \( x \):
\[ (x - 5)^3 = 1 - y^5 \]

### Step 4: Take the cube root
Next, take the cube root of both sides:
\[ x - 5 = (1 - y^5)^{1/3} \]

### Step 5: Solve for \( x \)
Finally, solve for \( x \) by adding 5 to both sides:
\[ x = 5 + (1 - y^5)^{1/3} \]

### Step 6: Express the inverse function
Since we have expressed \( x \) in terms of \( y \), we can write the inverse function:
\[ f^{-1}(y) = 5 + (1 - y^5)^{1/3} \]

### Step 7: Replace \( y \) with \( x \)
To express the inverse function in standard form, replace \( y \) with \( x \):
\[ f^{-1}(x) = 5 + (1 - x^5)^{1/3} \]

### Conclusion
Thus, the inverse of the function \( f(x) = [1 - (x - 5)^3]^{1/5} \) is indeed:
\[ f^{-1}(x) = 5 + (1 - x^5)^{1/3} \]