Question

If $R$ is the set of all real numbers and if $f:R-\left\{2\right\}\to R$ is defined by $f\left(x\right)=\frac{2+x}{2-x}$ for $x\in R-\left\{2\right\}$, then the range of $f$ is

• A. $R-\{-2\}$
• B. $R$
• C. $R-\left\{1\right\}$
• D. $R-\{-1\}$

Answer 1

Answer: (D)

$R-\{-1\}$

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

$\Rightarrow 2f\left(x\right)-xf\left(x\right)=2+x$

$\Rightarrow 2f\left(x\right)-2=xf\left(x\right)+x$

i.e. $x=\frac{2f\left(x\right)-2}{f\left(x\right)+1}$

$\Rightarrow f\left(x\right)\ne -1$

Hence the range of $f$ becomes $R-\{-1\}$