Question

If A = {1, 2, 3, 4}, f : R → R, f(x) = x² + 3x + 1 g : R → R, 8(x) = 2x – 3, then find

(i) (fog)(x)

(ii) (gof)(x)

(iii) (fof)(x)

(iv) (gog)(x)

Answer 1

Given,

f : R→ R, f(x) = x² + 3x + 1

g : R → R, g(x) = 2x – 3

(i) (fog)(x) = f{g(x)}

= f{2x – 3}

= (2x – 3)² + 3(2x – 3) + 1

= 4x² – 12x + 9 + 6x – 9 + 1

= 4x² – 6x + 1

(ii) (gof)(x) = g{f(x)}

= g(x² + 3x + 1)

= 2(x² + 3x + 1) – 3

= 2x² + 6x + 2 – 3

= 2x² + 6x – 1

(iii) (fof)(x) = 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)(x) = g{g(x)}

= g(2x – 3)

= 2(2x – 3) – 3

= 4x – 6 – 3

= 4x – 9