Question

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

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

Answer 1

Answer: (B)

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