Question:
Mark (√) against the correct answer in the following:
Let $A=R-\{3\}$ and $B=R-\{1\} .$ Then $f: A \rightarrow A: f(x)=\frac{(x-2)}{(x-3)}$ is
A. one - one and into
B. one - one and onto
C. many - one and into
D. many - one and onto
Solution:
$\mathrm{f}: \mathrm{A} \rightarrow \mathrm{A}: \mathrm{f}(\mathrm{x})=\frac{(\mathrm{x}-2)}{(\mathrm{x}-3)}$
In this function
$x=3$ and $y=1$ are the asymptotes of this curve and these are not included in the functions of the domain and range respectively therefore the function $f(x)$ is one one sice there are no different values of $x$ which has same value of $y$.
and the function has no value at $y=1$ here range $=$ codomain
$\therefore f(x)$ is onto