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Work Shown:
f(x) = x-2
g(x) = ax^2 + b
g(f(x)) = a( f(x) )^2 + b
g(f(x)) = a( x-2 )^2 + b
g(f(x)) = a(x^2-4x+4) + b
g(f(x)) = ax^2-4ax+4a + b
Compare this to g(f(x)) = 2x^2-8x+9, we see that the leading coefficient must be a = 2.
So,
g(f(x)) = ax^2-4ax+4a + b
g(f(x)) = 2x^2-4*2x+4*2 + b
g(f(x)) = 2x^2-8x+8 + b
Then compare the 8+b portion with the last term 9 in the given (gof)(x) equation. Solving 8+b = 9 leads to b = 1
g(f(x)) = 2x^2-8x+8 + b
g(f(x)) = 2x^2-8x+8 + 1
g(f(x)) = 2x^2-8x+9
Return back to the original form of g(x) to update 'a' and b
g(x) = ax^2 + b
g(x) = 2x^2 + 1
