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Answer:
f^-1(x) = <see attachment>
Step-by-step explanation:
If we write the function as ...
f(x) = ∛x +101/128x
Then the inverse function is the value of y when ...
x = f(y)
This resolves to a cubic in y, whose solution can be found by use of any of the various cubic solution formulas. You can find the cubic by ...
x = ∛y +by . . . . . . where b = 101/128
x -by = ∛y . . . . . . . subtract by
(x -by)³= y . . . . . . cube both sides
If we subtract the left side and expand the cube, we get ...
b³y³ -3b²xy² +(1 +3bx²)y -x³ = 0 . . . . . . . a cubic in y
There will be one real and two complex solutions to this cubic. The real solution can be written in the form shown in the attachment. That is, ...
y = f⁻¹(x) = [attachment expression]
A graph of it is shown in the second attachment. It is the reflection of f(x) across the line y = x, as an inverse function should be.
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Additional comment
The idea here is to show that it is possible to write an expression for the inverse function. It is not pretty, and it is somewhat tedious to evaluate.
