Given f(x)=x^2-x and g(x)=ax+b what are a and b if fog=gof .

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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We'll compose the functions f and g.

fog(x) = f(g(x)) = [g(x)]^2 - g(x)

f(g(x)) = (ax+b)^2 - ax - b

gof(x) = g(f(x)) = a*f(x) + b

g(f(x)) = a*(x^2 - x) + b

To determine a and b, we'll impose the constraint given by enunciation:

fog=gof

(ax+b)^2 - ax - b = a*(x^2 - x) + b

We'll expand the square from the left side and we'll remove the brackets from the right side:

a^2*x^2 + 2axb + b^2 - ax - b = ax^2 - ax + b

We'll move all terms to one side:

x^2(a^2 - a) + x(2ab - a + a) + b^2 - 2b = 0

Comparing, we'll get:

a^2 - a = 0

a(a-1) = 0

a = 0 and a-1 = 0 => a = 1

2abĀ  = 0 => b = 0

b^2 - 2b = 0

b(b - 2) = 0

b = 0 and b = 2

The possible values are: a = {0; 1} and b = {0 ; 2}.

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