Functions f , g : R → R are defined,
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

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