f(y*f(x)-x)=f(x)*f(y)+2x

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Solution for f(y*f(x)-x)=f(x)*f(y)+2x equation: 


Simplifying
f(y * f(x) + -1x) = f(x) * f(y) + 2x

Multiply y * f
f(fy * x + -1x) = f(x) * f(y) + 2x

Multiply fy * x
f(fxy + -1x) = f(x) * f(y) + 2x
(fxy * f + -1x * f) = f(x) * f(y) + 2x

Reorder the terms:
(-1fx + f2xy) = f(x) * f(y) + 2x
(-1fx + f2xy) = f(x) * f(y) + 2x

Multiply f * x
-1fx + f2xy = fx * f * y + 2x

Multiply fx * f
-1fx + f2xy = f2x * y + 2x

Multiply f2x * y
-1fx + f2xy = f2xy + 2x

Add '-1f2xy' to each side of the equation.
-1fx + f2xy + -1f2xy = f2xy + -1f2xy + 2x

Combine like terms: f2xy + -1f2xy = 0
-1fx + 0 = f2xy + -1f2xy + 2x
-1fx = f2xy + -1f2xy + 2x

Combine like terms: f2xy + -1f2xy = 0
-1fx = 0 + 2x
-1fx = 2x

Solving
-1fx = 2x

Solving for variable 'f'.

Move all terms containing f to the left, all other terms to the right.

Divide each side by '-1x'.
f = -2

Simplifying
f = -2

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