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