Question:
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
Solution:
Given, f(x) = x2 + 3x + 1, g (x) = 2x – 3
(i) fog = f(g(x))
= f(2x – 3)
= (2x – 3)2 + 3(2x – 3) + 1
= 4x2 + 9 – 12x + 6x – 9 + 1
= 4x2 – 6x + 1
(ii) gof = g(f(x))
= g(x2 + 3x + 1)
= 2(x2 + 3x + 1) – 3
= 2x2 + 6x – 1
(iii) 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
(iv) gog = g(g(x))
= g(2x – 3)
= 2(2x – 3) – 3
= 4x – 6 – 3
= 4x – 9