The real valued function $f: \mathrm{R} \rightarrow\left[\frac{5}{2}, \infty\right)$ defined by $f(x)=|2 x+1|+1|x-2|$ is

• A

  One-one function but not onto
• Correct answer
  B

  Onto function but not one-one
• C

  Bijection
• D

  Neither one-one function nor onto

Solution

$f: \mathbb{R} \rightarrow\left[\frac{5}{2}, \infty\right), f(x)=|2 x+1|+|x-2|$
$\rightarrow f(x)=\left\{\begin{array}{lc}-3 x+1, & x \lt \frac{-1}{2} \\ x+3, & \frac{-1}{2} \lt x \lt 2 \\ 3 x-1, & x\gt2\end{array}\right.$
Clearly, any horizontal line will cut the graph atleast twice. $\therefore f(x)$ is not one-one and clearly, $f(x)$ is onto.

Hence, the correct answer is option: 'B'.