Functions f, g : R → R are defined, respectively, by f (x) = x2 + 3x + 1, g (x) = 2x – 3, find
(i) f o g (ii) g o f (iii) f o f (iv) g o g
Given that, f (x) = x2 + 3x + 1, g (x) = 2x – 3
(i) f o g
fog = f(g(x)) = f(2x-3)
= (2x-3)2 + 3(2x-3) + 1
= (4x2-12x+9) + 6x – 9 +1
= 4x2 - 6x + 1
∴ fog = 4x2 - 6x + 1
(ii) g o f
gof = g(f(x)) = g(x2 + 3x + 1)
= 2(x2 + 3x + 1) – 3
= 2x2 + 6x + 2 – 3
= 2x2 + 6x – 1
∴ gof = 2x2 + 6x – 1
(iii) f o f
fof = f(f(x)) = f(x2 + 3x + 1)
= (x2 + 3x + 1)2 + 3(x2 + 3x + 1) + 1
= x4+9x2+1+6x3+6x+2x2+3x2+9x+3+1
= x4+6x3+14x2+15x+5
∴ fof = x4+6x3+14x2+15x+5
(iv) g o g
gog = g(g(x)) = g(2x-3)
= 2(2x-3) – 3
= 4x-6-3
= 4x-9
∴ gog = 4x-9