Question

Find the range of following quadratic expression.
$f\left(x\right)=-x^2+2x+3$ $\forall x\in R$.

Answer 1

Let, $F\left(x\right)=y$

So, $y=-x^2+2x-3$

$y=-(x^2-2x-3)$

$y=-(x^2-2x+1-4)$

$y=-(x^2-2x+1)-(-4)$

$y=-(x-1{)}^2+4$

as we know, $(x-1{)}^2\ge 0$

So, $-(x-1{)}^2\le 0$

now add $4$ both side.

So, $-(x-1{)}^2+4\le 0+4$

So, $-(x-1{)}^2+4=y.$

$y\le 4$

So, the range of $y(i.eF(x)$ is $(-\infty ,4]$

$\therefore -\infty <y\le 4$