Question

A function $f\left(x\right)=\sqrt{1-2x}+x$ is defined from $D_1\to D_2$ and is onto. If the set $D_1$ is its complete domain then the set $D_2$ is

• A. $(-\infty ,\frac{1}{2}]$
• B. $(-\infty ,2)$
• C. $(-\infty ,1)$
• D. $(-\infty ,1]$

Answer 1

The correct option is D $(-\infty ,1]$

We know, $1-2x\ge 0$
$1\ge 2x$
$x\le \frac{1}{2}$
Hence domain is $(-\infty ,\frac{1}{2}]$
Now
$\frac{df\left(x\right)}{dx}$
$=\frac{-1}{\sqrt{1-2x}}+1$
$=0$ for maxima minima.
$\sqrt{1-2x}=1$
$1-2x=1$
$\therefore x=0$
Hence $f\left(0\right)=1$ which is the maxima.
Thus range is $(-\infty ,1]$