Given: f(x)=x+3 g(x)=(ax^2)+bx+c g(f(x))=(2x^2)+7x+6

Question

Find a+b+c PLEASE SHOW WORK?

Answer 1

The functions are f(x) = x +3 and g(x) = ax²+bx+c

g(f(x)) = 2x²+7x+6

Now we need to find the composite function g(f(x))

g(f(x)) = g(x+3)

g(f(x)) = a(x+3)² + b(x+3) + c

g(f(x)) = a(x² + 9 + 6x) + bx + 3b + c

g(f(x)) = ax² + 9a + 6ax + bx + 3b + c

g(f(x)) = ax²+x(b+6a) + (9a+3b+c)

Now compare the above equation g(f(x)) = 2x²+7x+6.

Compare x² coefficient

a = 2

Compare x coefficient

b + 6a = 7

b + 6*2 = 7

b + 12 = 7

b = 7 -12

b = -5

Compare constants

9a + 3b + c = 6

9*2 + 3*(-5) + c = 6

18 - 15 + c = 6

3 + c = 6

c = 6 - 3

c = 3

Then a +b + c = 2 - 5 + 3 = 0

Therefore a+b+c=0.