Question

Let a function $f$ be defined from $R\to R$ by $f\left(x\right)=\{\begin{array}{cc}x+m,& x\le 1\\ 2mx-1,& x>1\end{array}$
If the function $f$ is surjective, then $m\in$

• A. $[-2,2]$
• B. $R-\{-2,1,2\}$
• C. $R-\left\{1\right\}$
• D. $(0,2]$

Answer 1

The correct option is D $(0,2]$

Given : $f:R\to R$ is surjective(onto), $\Rightarrow$ Range $=R$
Now,
When $x\le 1:$
$x+m\le m+1$
$\Rightarrow$ Range $\in (-\infty ,m+1]\cdots \left(i\right)$

When $x>1,m>0:$
$\Rightarrow mx>m\Rightarrow 2mx-1>2m-1$
$\Rightarrow$ Range $\in (2m-1,\infty )\cdots \left(ii\right)$

When $x>1,m<0:$
$\Rightarrow mx<m\Rightarrow 2mx-1<2m-1$
$\Rightarrow$ Range $\in (-\infty ,2m-1)\cdots \left(iii\right)$

From $\left(i\right)$ and $\left(ii\right):m>0$

For Range to be $R,2m-1\le m+1$
$\Rightarrow 0<m\le 2\cdots \left(A\right)$

and from $\left(i\right)$ and $\left(iii\right):$


Range is either $(-\infty ,m+1]$ or $(-\infty ,2m-1)$
$\Rightarrow$ Range $\notin R,$ for any value of $m.$
$\therefore$ only possible set of $m$ for which $f$ to be surjective is $\left(A\right):(0,2]$