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