The range of the function f(x)= _e (3 x^2+4 ) is - Vidyadip

MCQ

The range of the function $f(x)=\log _e\left(3 x^2+4\right)$ is

A
$\left[\log _e 2, \infty\right]$
B
$\left[\log _e 3, \infty\right)$
C
$\left[2 \log _e 3, \infty\right)$
✓
$\left[2 \log _e 2, \infty\right)$

✓

Answer

Correct option: D.

$\left[2 \log _e 2, \infty\right)$

(D)
Let $y=\log _e\left(3 x^2+4\right)$
$\Rightarrow 3 x^2+4= e ^y$
$\Rightarrow x^2=\frac{ e ^y- 4}{3}$
Since, $x^2 \geq 0$
$\therefore \frac{ e ^y-4}{3} \geq 0 \Rightarrow e ^y-4 \geq 0 \Rightarrow y \geq \log _{ e } 4$
$\Rightarrow y \geq 2 \log _c 2$
So, range $=\left[2 \log _{ e } 2, \infty\right)$

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