Question

Let a function $f:(0,\infty )\to (0,\infty )$ be defined by $f(x)=|1-\frac{1}{x}|$ . Then $f$ is :

• A. Injective only
• B. Not injective but it is surjective
• C. Both injective as well as surjective
• D. Neither injective nor surjective

Answer 1

Answer: (B)

$f\left(x\right)=\left|1-\frac{1}{x}\right|=\frac{|x-1|}{x}=\{\begin{array}{ll}\frac{1-x}{x}& 0<x\le 1\\ \frac{x-1}{x}& x\ge 1\end{array}$
$\Rightarrow f(x)$ is not injective
but range of function is $(0,\infty )$
If co-domain is $(0,\infty )$, then $f(x)$ will be surjective