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This question already has an answer here:

I want to calculate the inverse of

f[x_] := 1/2 - (x (4 x^2 - 9))/12 /; -1/2 <= x <= 1/2

f[x] is monotonic inside $-0.5 < x <= 0.5$, but I don't know how to calculate its inverse.

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marked as duplicate by Artes, bobthechemist, m_goldberg, Michael E2, Sjoerd C. de Vries Feb 4 '14 at 6:22

This question was marked as an exact duplicate of an existing question.

1  
does it help? – Wojciech Feb 3 '14 at 9:55
    
This is answered in the documentation for InverseFunction – Michael E2 Feb 4 '14 at 4:03

The inverse you are looking for seems to be

invF3[c_] := (3^(2/3)*(1 - I*Sqrt[3]))/
    (4*(-2 + 4*c + Sqrt[1 - 16*c + 16*c^2])^(1/3)) + 
   (1/4)*3^(1/3)*(1 + I*Sqrt[3])*(-2 + 4*c + Sqrt[1 - 16*c + 16*c^2])^(1/3)

You can figure this out by doing

Solve[c == 1/2 - (x (4 x^2 - 9))/12, x]

and then trying out the solutions. Anyway we have

invF3[f[1/4]] // FullSimplify
invF3[f[0]] // Simplify
f[invF3@0] // Simplify

1/4
0
0

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