Functions $f,g:R \to R$ are defined, respectively, by $f(x) = {x^2} + 3x + 1$ $g(x) = 2x - 3$, find
(i) fog
(ii) gof
(iii) fof
(iv) gog

It is given that,, $f(x) = {x^2} + 3x + 1,g(x) = 2x - 3$

(i) $fog = f\{ g(x)\} = f(2x - 3)$

$= {(2x - 3)^2} + 3(2x - 3) + 1$

$= 4{x^2} + 9 - 12x + 6x - 9 + 1 = 4{x^2} - 6x + 1$

(ii) $gof = g\{ f(x)\} = g\left( {{x^2} + 3x + 1} \right)$

$= 2\left( {{x^2} + 3x + 1} \right) - 3$

$= 2{x^2} + 6x + 2 - 3 = 2{x^2} + 6x - 1$

(iii) $fof = f\{ f(x)\} = f\left( {{x^2} + 3x + 1} \right)$

$= {\left( {{x^2} + 3x + 1} \right)^2} + 3\left( {{x^2} + 3x + 1} \right) + 1$

$= {x^4} + 9{x^2} + 1 + 6{x^3} + 6x + 2{x^2} + 3{x^2} + 9x + 3 + 1$

$= {x^4} + 6{x^3} + 14{x^2} + 15x + 5$

(iv) $gog = g\{ g(x)\} = g(2x - 3)$

$= 2(2x - 3) - 3$

$= 4x - 6 - 3 = 4x - 9$