Question

Let $f:D\to \{1,3,9,19\}$ be a function defined as $f\left(x\right)=2x^2+1$, where $D$ is the domain of $f$. If range and co-domain of the function $f$ are equal, then the maximum number of elements in set $D$ can be

Answer 1

$f:D\to \{1,3,9,19\}$ and range and co-domain of the function are equal.
Here, each and every element of co-domain should have at least one pre-image in $D$.

For $f\left(x\right)=1$
$2x^2+1=1$
$\Rightarrow 2x^2=0$
$\Rightarrow x=0$

For $f\left(x\right)=3$
$2x^2+1=3$
$\Rightarrow x=-1,1$

For $f\left(x\right)=9$
$2x^2+1=9$
$\Rightarrow x=-2,2$

For $f\left(x\right)=19$
$2x^2+1=19$
$\Rightarrow x=-3,3$

So, $D$ with maximum number of elements can be, $D=\{-3,-2,-1,0,1,2,3\}$
$\therefore$ Maximum number of elements can be $7.$