Does order of functions really matter?

Algebra Level pending

Suppose f ( x ) = x 2 f(x) = x^2 and g ( x ) = x + 2 g(x) = x +2 . If x = − a b x = -\frac{a}{b} for which f g ( x ) = g f ( x ) fg(x) = gf(x) where a and b are positive coprime integers, a + b = ? a+b =?


The answer is 3.

Noel Lo by Noel Lo , 24, Singapore

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1 solution

Noel Lo
Apr 12, 2015

f g ( x ) = g f ( x ) fg(x) =gf(x)

f ( x + 2 ) = g ( x 2 ) f(x+2) = g(x^2)

( x + 2 ) 2 = x 2 + 2 (x+2)^2 = x^2 + 2

x 2 + 4 x + 4 = x 2 + 2 x^2 +4x+4 = x^2 + 2

4 x = − 2 4x = -2

x = − 1 2 x = -\frac{1}{2} such that a + b = 1 + 2 = 3 a+ b = 1+2 =\boxed{3}

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