Given that f(x) = x - 2 and (gof)(x) = 2x^2 - 8x + 9, find g(x).

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

answer: finding the extract measurements and knowing the points

step-by-step explanation:

a)(x-2)^2+(y-1)^2=9

step-by-step explanation:

(picture explains)