Question

Let a function f defined from $R\to R$ as $f\left(x\right)=\{\begin{array}{cc}x+p^2,& x\le 2\\ 2px+5,& x>2\end{array}$.
If the function is surjective, then sum of all possible integral value of p in [-100, 100] is

Answer 1

We have,
$f\left(x\right)=\{\begin{array}{cc}x+p^2,& x\le 2\\ 2px+5,& x>2\end{array}$.
Case I: $p>0$
Range of f(x) will be = $(-\infty ,2]+p^2=(-\infty ,2+p^2]forallx\le 2$ and $2p(2,\infty )+5=(4p+5,\infty )forallx>2$
Since, the given function f(x) is an onto function.
$\therefore 2+p^2\ge 4p+5\Rightarrow p^2-4p-3\ge 0\Rightarrow p\ge 2+\sqrt{7}orp\le 2-\sqrt{7}\text{(which is not possible as p > 0}\therefore p\ge 2+\sqrt{7}>4\text{Hence,p can take values 5, 6, 7,....., 100.}\text{Required sum for p > 0}=5+6+7+....+100=5050-10=5040$
Case II: $p<0$
Range of f(x) will be = $(-\infty ,2]+p^2=(-\infty ,2+p^2]forallx\le 2$ and $2p(2,\infty )+5=(-\infty ,4p+5)forallx>2$
Function cannot take all the values in the set R and hence, will not be an onto function.
Therefore, p < 0 is not possible.
Required Sum = 5040