If a function $\mathrm{f}: \mathbf{R}-\{l\} \mathbf{R}-\{m\}$ defined by…
If a function $\mathrm{f}: \mathbf{R}-\{l\} \mathbf{R}-\{m\}$ defined by $\mathrm{f}(\mathrm{x})=\frac{x+3}{x-2}$ is a bijection, then $3 l-2 m=$
10
12
8
14
Solution
$f(x)=\frac{x+3}{x-2}$
$\because \quad f(x)$ is not defined for $x=2$
i.e. domain of $f(x)$ is $\mathrm{R}-\{2\}$
$\therefore l=2$
Now, $y=\frac{x+3}{x-2}$
$\begin{aligned} & x y-2 y=x+3 \\ & x(y-1)=2 y+3 \\ & x=\frac{2 y+3}{y-1} \\ & \because y \text { can take any value except } \\ & y=1 \\ & \therefore \text { co-domain }=\mathrm{R}-\{1\}\end{aligned}$
$m=1$
$3 l+2 m=3(2)+2(1)=8$