Question

Function $f\left(x\right)=\frac{x}{1+x}$ defined from $[0,\infty )\to [0,\infty ),f\left(x\right)$ 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

The correct option is A one-one and onto

$f\left(x\right)=\frac{x}{1+x}$

Llet $x_1,x_2\in [0,\infty )$

$f\left(x_1\right)=f\left(x_2\right)$

$\Rightarrow \frac{x_1}{1+x_1}=\frac{x_2}{1+x_2}\Rightarrow x_1=x_2$

Therefore, function $f$ is one-one

Let $f\left(x\right)=y$

$y=\frac{x}{1+x}$

$y+yx=x$

$\Rightarrow x(1-y)=y$

$x=\frac{y}{1-y}$

$f\left(x\right)=\frac{x}{1+x}$

$f\left(\frac{y}{1-y}\right)=\frac{\frac{y}{1-y}}{1+\frac{y}{1-y}}=y$

$f\left(x\right)=y$

$\Rightarrow$ Function $f$ is onto.