Question

Find the domain and the range of the real function, $f\left(x\right)=\frac{3-x}{x-3}$

Answer 1

We have, $f\left(x\right)=\frac{3-x}{x-3}$

Clearly, $f\left(x\right)$ is defined for all real values of $x$ for which $x-3\ne 0$, i.e., $x\ne 3$.

$\therefore \text{dom (f)}=R-\left\{3\right\}$

Let $y=f\left(x\right)$. Then,

$y=\frac{3-x}{x-3}\Rightarrow y=-1$, when $x-3\ne 0$

$\Rightarrow y=-1$, when $x\ne 3$

$\therefore$ range (f) = {-1}

Hence, $\text{dom (f)}=R-\left\{3\right\}$ and range (f) = {-1}.