Question

The range of the function $f\left(x\right)=\frac{1+x}{1-x},x\ne 1$ is:

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

Answer 1

The correct option is B $R-\{-1\}$

Let $f\left(x\right)=y$

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

$\Rightarrow (1-x)y=1+x$

$\Rightarrow x(y+1)=y-1$

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

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

It is evident that $f^{-1}\left(x\right)$ exists for all $R$ except $R=-1$.

$\therefore$ range of $f\left(x\right)$ is $R-\{-1\}$