Composition of function

Question

Let f(x) = x^2 + 4x – 12 and f(g(x)) = x^2 – 10x + 9. Determine g(x)

Answer 1

f(x)=x^2+4x-12

f(g(x))= x^2-10x+9

let g(x)=ax+b, solve for a and b

f(ax+b)= (ax+b)^2 +4(ax+b) -12

= a^2x^2 +2abx +b^2 +4ax+4b-12=x^2-10x +9

set coefficients on left and right sides equal

a=1 for x^2

2ab+4a=-10=2(1)b+4(1) for x

2b=-14

b= -7

g(x)= ax+b=(1)x-7

g(x)=x-7

check the answer

b^2+4b-12=9 for the constant term

49-28-12 =9

9=9

f(g(x)=f(x-7)= (x-7)^2+4(x-7)-12

= x^2-14x+49 +4x-28 -12

= x^2 -10x +9