Question

The domain and range of the function $f(x)$ given by $f\left(x\right)=\frac{x-2}{3-x}$ respectively are

• A. $R,R-\left\{1\right\}$
• B. $R-\left\{3\right\},R-\left\{-1\right\}$
• C. $R-\left\{1\right\},R-\left\{3\right\}$
• D. None of these

Answer 1

Answer: (B)

We have, $f\left(x\right)=\frac{x-2}{3-x}$
Clearly, $f\left(x\right)$ is defined for all $x$ satisfying $3-x\ne 0$ i.e. $x\ne 3$.
Hence, Domain $\left(f\right)=R-\left\{3\right\}$.
Let $y=f\left(x\right)$, i.e. $y=\frac{x-2}{3-x}$
$\Rightarrow 3y-xy=x-2$
$\Rightarrow x\left(y+1\right)=3y+2$
$\Rightarrow x=\frac{3y+2}{y+1}$
Clearly, $x$ assumes real values for all $y$ except $y+1=0$ i.e. $y=-1$.
Hence, Range $\left(f\right)=R-\left\{-1\right\}$.