Functions Ques 16
- Let $A=\{x \in R: x$ is not a positive integer $\}$. Define a function $f: A \rightarrow R$ as $f(x)=\frac{2 x}{x-1}$, then $f$ is
(2019 Main, 9 Jan II)
(a) injective but not surjective
(b) not injective
(c) surjective but not injective
(d) neither injective nor surjective
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Answer:
Correct Answer: 16.(a)
Solution: (a) We have a function $f: A \rightarrow R$ defined as, $f(x)=\frac{2 x}{x-1}$
One-one Let $x_1, x_2 \in A$ such that
$\begin{aligned} f\left(x_1\right) & =f\left(x_2\right) \\ \Rightarrow \quad \frac{2 x_1}{x_1-1} & =\frac{2 x_2}{x_2-1}\end{aligned}$
$\Rightarrow \quad 2 x_1 x_2-2 x_1=2 x_1 x_2-2 x_2$ $\Rightarrow \quad x_1=x_2$
Thus, $f\left(x_1\right)=f\left(x_2\right)$ has only one solution, $x_1=x_2$
$\therefore \quad f(x)$ is one-one (injective)
Onto Let $x=2$, then $f(2)=\frac{2 \times 2}{2-1}=4$
But $x=2$ is not in the domain, and $f(x)$ is one-one function
$\therefore \quad f(x)$ can never be $4$ .
Similarly, $f(x)$ can not take many values.
Hence, $f(x)$ is into (not surjective).
$\therefore \quad f(x)$ is injective but not surjective.
BETA
