Let g(x) = 2x-3 and h(x) = 2x^2 + 4x -8. Find f(x) such that fog (composition) = h

Question

the answer for the question is  1/2x^2 +5x+5/2 im just not sure the procedure. Thanx

Answer 1

Let g(x) = 2x-3 and h(x) = 2x^2 + 4x -8. Find f(x) such that fog (composition) = h

To find f() s.t. fog = h

g^2 = 4x^2 – 12x + 9

Now,

h = 2x^2 + 4x + 8

h = (4x^2 + 8x + 16)/2

h = (4x^2 – 12x + 20x + 9 + 7)/2

h = (4x^2 – 12x + 9)/2 + (20x + 7)/2

h = g^2/2 + 10x + 3 ½

h = g^2/2 + 10x – 15 + 15 + 3 ½

h = g^2/2 + 5(2x – 3) + 18 ½

h = g^2/2 + 5g + 18 ½

And, fog = h, i.e.

f(g) = g^2/2 + 5g + 18 ½

f(x) = (x^2 + 10x + 37)/2