Let $f: R \rightarrow R: f(x)=x^{2}+2$ and $g: R \rightarrow R: g(x)=\frac{x}{x-1}, x \neq 1$. find $f$ o $g$ and $g$ o $f$ and hence find (f o $g$ ) (2)
and (g o f) $(-3)$
To find: $f \circ g, g \circ f,(f \circ g)(2)$ and $(g \circ f)(-3)$
Formula used: (i) $f \circ g=f(g(x))$
(ii) $g \circ f=g(f(x))$
Given: (i) $f: R \rightarrow R: f(x)=x^{2}+2$
(ii) $g: R \rightarrow R: g(x)=\frac{x}{x-1}, x \neq 1$
f o g = f(g(x))
$\Rightarrow f\left(\frac{x}{x-1}\right)$
$\Rightarrow\left(\frac{x}{x-1}\right)^{2}+2$
Ans $) \Rightarrow \frac{(x)^{2}}{(x-1)^{2}}+2$
fog $(2)=\frac{(2)^{2}}{(2-1)^{2}}+2$
$=\frac{4}{1}+2$
Ans) = 6
g o f = g(f(x))
$\Rightarrow g\left(x^{2}+2\right)$
$\Rightarrow \frac{x^{2}+2}{x^{2}+2-1}$
Ans $) \Rightarrow \frac{x^{2}+2}{x^{2}+1}$
$(g \circ f)(-3)=\frac{-3^{2}+2}{-3^{2}+1}$
$=\frac{9+2}{9+1}$
Ans) $=\frac{11}{10}$