Question

The range of the function $f\left(x\right)=2\sqrt{3x^2−4x+5}$ is

• A. $\left[−∈fty,2\sqrt{\frac{11}{3}}\right]$
• B. $\left(−∈fty,2\sqrt{\frac{11}{3}}\right)$
• C. $\left[2\sqrt{\frac{11}{3}},∞\right)$
• D. $\left(2\sqrt{\frac{11}{3}},∈fty\right)$

Answer 1

Answer: (C)

$f\left(x\right)$ is defined if $3x^2−4x+5â‰\yen 0$
$⇒3\left[x^2−\frac{4}{3}x+\frac{5}{3}\right]â‰\yen 0⇒\left[{\left(x−\frac{2}{3}\right)}^2+\frac{11}{9}\right]â‰\yen 0$
Which is true for all real $x$
$∴$ Domain $\left(f\right)=\left(−∈fty,∈fty\right)$
Let, $y=2\sqrt{3x^2−4x+5}$
$⇒\frac{y^2}{4}=3x^2−4x+5i.e.3x^2−4x+\left(5−\frac{y^2}{4}\right)=0$
For $x$ to be real, $16−12\left(5−\frac{y^2}{4}\right)â‰\yen 0$
$⇒\frac{y}{2}â‰\yen \sqrt{\frac{11}{3}}$
$∴$ Range of $y=\left[2\sqrt{\frac{11}{3}},∞\right)$