Question

f(x)=3x^2 -1 & g(x)=x^2 +x+1. find fog and gof???​

Answer 1

Stepwise solution:

Given

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

and

$g(x) = {x}^{2} + x + 1$

To find:

$fog \: \text{and} \: gof$

$fog(x) = f(g(x)) \\ = f( {x}^{2} + x + 1) \\ = 3( {{x}^{2} + x + 1)}^{2} - 1 \\ = 3( {x}^{4} + {x}^{2} + 1 + 2 {x}^{3} + 2x + 2 {x}^{2} ) - 1 \\ = 3({x}^{4} + {x}^{2} + 2 {x}^{3} + 2x + 2 {x}^{2}) + 2.$

Similarly,

$gof(x) = g(f(x)) \\ = g(3 {x}^{2} - 1) \\ = {(3 {x}^{2} - 1)}^{2} + 3 {x}^{2} - 1 + 1 \\ = 9 {x}^{4} + 1 - 6 {x}^{2} + 3 {x}^{2} \\ = 9 {x}^{4} - 3 {x}^{2} + 1.$