Find the Inverse f(x)=4x^3-2x

Problem

ƒ(x)=4*x3−2*x

Solution

1. Replace ƒ(x) with y to set up the equation for the inverse.

y=4*x3−2*x

2. Swap the variables x and y to represent the inverse relationship.

x=4*y3−2*y

3. Analyze the equation to solve for y This is a cubic equation in the form 4*y3−2*y−x=0

4. Apply Cardano's formula for a cubic equation of the form a*y3+b*y2+c*y+d=0 where a=4 b=0 c=−2 and d=−x

5. Calculate the discriminant for the cubic to determine the nature of the roots. For certain values of x there may be multiple real roots, meaning the function is not one-to-one over its entire domain.

6. Express the real root using the cubic formula for y

y=√(3,x/8+√(,(x2)/64−1/27))+√(3,x/8−√(,(x2)/64−1/27))

Final Answer

ƒ(x)(−1)=√(3,x/8+√(,(x2)/64−8/216))+√(3,x/8−√(,(x2)/64−8/216))

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