Question

The function $f:[0,\infty )\to [0,\infty )$ defined by $f(x)=\frac{2x}{1+2x}$ is

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

Answer 1

Answer: (B)

Given, $f:[0,\infty )\to [0,\infty )$
and $f(x)=\frac{2x}{1+2x}$
Let, $y=\frac{2x}{1+2x}$
$\Rightarrow$ $y+2xy=2x$
$\Rightarrow$ $y=2x(1-y)$
$\Rightarrow$ $x=\frac{y}{2(1-y)}$
Here, $y\in R\stackrel{~}{}\{1\}.$
So, the range of $f(x)$ is $R\stackrel{~}{}\{1\}.$
Since, Range $\ne$ codomain
So, the given function is on to. For one-one,
$f(x_1)=f(x_2)$
$\Rightarrow$ $\frac{2x_1}{1+x_1}=\frac{2x_2}{1+x_2}$
$\Rightarrow$ $x_1+x_1{x}_2=x_2+x_1{x}_2$
$\Rightarrow$ $x_1=x_2$
So, the above function is one-one.