Question

If $f:[1,\infty )\to [0,\infty )andf\left(x\right)=\frac{x}{1+x}$ then f is

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

Answer 1

The correct option is A one-one and into

Given that $f:[0,\infty )\to [0,\infty )$
$s.t.f\left(x\right)=\frac{x}{x+1}$
then $f^{\prime }\left(x\right)$=$\frac{1+x-x}{(1+x^2)}=\frac{1}{(1+x{)}^2}$ >0,$\forall x$
$\therefore$ f is an increasing function $\Rightarrow$ is one-one.
Also $D_f=[0,\infty )$
And for range let $\frac{x}{x+1}=y\Rightarrow x=\frac{y}{y+1}$