Question

The range of $f\left(x\right)=\frac{x+2}{x-3}$ is

• A. $R-\left\{3\right\}$
• B. $R-\left\{1\right\}$
• C. $(-\infty ,-3)\cup (-2,\infty )$
• D. $(0,\infty )$

Answer 1

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

$f$ is defined for all $x$ except where $x-3=0$ i.e., $x=3$
$\therefore D\left(f\right)=R-\left\{3\right\}$

Let $y=\frac{x+2}{x-3}$
$\Rightarrow xy-3y=x+2$
$\Rightarrow xy-x=3y+2$
$\Rightarrow x=\frac{3y+2}{y-1}$
$x$ is defined for all $y\in R-\left\{1\right\}$
$\therefore R\left(f\right)=R-\left\{1\right\}$