Q. 25

# 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

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