(f+g)(X)=F(x)+G(x)

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Solution for (f+g)(X)=F(x)+G(x) equation: 


Simplifying
(f + g)(X) = F(x) + G(x)

Reorder the terms for easier multiplication:
X(f + g) = F(x) + G(x)
(f * X + g * X) = F(x) + G(x)
(fX + gX) = F(x) + G(x)

Multiply F * x
fX + gX = xF + G(x)

Multiply G * x
fX + gX = xF + xG

Solving
fX + gX = xF + xG

Solving for variable 'f'.

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

Add '-1gX' to each side of the equation.
fX + gX + -1gX = xF + -1gX + xG

Combine like terms: gX + -1gX = 0
fX + 0 = xF + -1gX + xG
fX = xF + -1gX + xG

Reorder the terms:
fX = -1gX + xF + xG

Divide each side by 'X'.
f = -1g + xFX-1 + xGX-1

Simplifying
f = -1g + xFX-1 + xGX-1

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