Question

Let $f$ : $[0$, $\infty) \to [0$, $2]$ be defined by $f \left(x\right)=\frac{2x}{1+x}$, then $f$ is

• A. one-one but not onto
• B. onto but not one-one
• C. both one-one and onto
• D. neither one-one nor onto

Answer 1

Answer: (A)

We have, $y=\frac{2x}{x+1}$
$\Rightarrow 2x = yx + y$
$\Rightarrow x=\frac{y}{2-y}$
Now, $x \in [0$, $\infty)$
$\Rightarrow 0 \le y < 2$
$\Rightarrow $ Range of $f(x) = [0$, $2)$
Since range $\subset$ co-domain
$\Rightarrow f$ is into. Clearly, $f$ is one-one also.