Question

If the function $f: R \rightarrow R$ be given by $f(x)=x^2+2$ and $g: R \rightarrow R$ be given by $g(x)=\frac{x}{x-1}$, then find $g^{-1}(x)$ and $\mathrm{fog}(x)$

(a) $\frac{x}{x-1}, \frac{3 x^2+4 x-1}{(x-1)^2}$

(b) $\frac{x}{x-1}, \frac{3 x^2-4 x+1}{(x-1)^2}$

(c) $\frac{x}{x+1}, \frac{3 x^2-4 x+1}{(x-1)}$

(d) Nome of these

Answer 1

$\text { Let } y=g(x)=\frac{x}{x-1}$

$y(x-1)=x$

$x=\frac{y}{y-1}$

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

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

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

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

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

So, The correct option of this question will be (D).