Question

If domain of the function $f\left(x\right)=x^2-6x+7$ is $(-\infty ,\infty )$, then its range is

• A. $[-2,3]$
• B. $(-\infty ,-2)$
• C. $(-\infty ,\infty )$
• D. $[-2,\infty )$

Answer 1

Answer: (D)

$[-2,\infty )$

Let $y=x^2-6x+7$
$\Rightarrow x^2-6x+7-y=0$
On comparing with $a{x}^2+bx+c=0$, we get
$a=1$, $b=-6$ and $c=(7-y)$
Now,
$x=\frac{-b\pm \sqrt{b^2-4ac}}{a}$
$=\frac{6\pm \sqrt{36-4(7-y)}}{2}$
$=\frac{6\pm \sqrt{36-28+4y}}{2}$
$=\frac{6\pm \sqrt{4y+8}}{2}$
$=\frac{6\pm 2\sqrt{y+2}}{2}$
$=3\pm \sqrt{y+2}$
$f\left(x\right)$ is defined only when
$y+2\ge 0\Rightarrow y\ge -2$
$\therefore$ Range of $f\left(x\right)=[-2,\infty )$.