Question

If the function $f:R\to R$ be given by $f\left(x\right)=x^2+2$ and $g:R\to R$ be given by $g\left(x\right)=\frac{x}{x-1},x\ne 1$, find fog and gof and hence find $fog\left(2\right)$ and $gof(-3)$

Answer 1

$f:R\to R$

$f\left(x\right)=x^2+2$

$g:R\to R$

$g\left(x\right)=\frac{x}{x-1}$

$fog\left(x\right)=f\left(g\right(x\left)\right)$

$=f\left(\frac{x}{x-1}\right)$

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

$=\frac{x^2}{x^2-2x+1}+2$

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

$=\frac{x^2+2x^2-4x+2}{x^2-2x+1}$

$=\frac{3x^2-4x+2}{x^2-2x+1}$

$fog\left(2\right)=\frac{3\times 2^2-4\times 2+2}{2^2-2\times 2+1}$

$=\frac{3\times 4-8+2}{4-4+1}$

$=\frac{12-6}{1}$

$=6$

$gof\left(x\right)=g\left(f\right(x\left)\right)$

$=g(x^2+2)$

$=\frac{x^2+2}{x^2+2-1}$

$=\frac{x^2+2}{x^2+1}$

$gof(-3)=\frac{(-3{)}^2+2}{(-3{)}^2+1}$

$=\frac{9+2}{9+1}$

$=\frac{11}{10}$