First slide

Functions (XII)

Question

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

Moderate

A

$[- \infty, \sqrt{\frac{11}{3}}]$
B

$(- \infty, \sqrt{\frac{11}{3}})$
C

$[\sqrt{\frac{11}{3}}, \infty)$
D

$(\sqrt{\frac{11}{3}}, \infty)$

Solution

f (x) is defined if 3x2 – 4x + 5 ≥ 0
$\Rightarrow 3 [x^{2} - \frac{4}{3} x + \frac{5}{3}] \geq 0 \Rightarrow 3 [{(x - \frac{2}{3})}^{2} + \frac{11}{9}] \geq 0$
which is true for all real x
$∴$ Domain ( f ) = (– ∞, ∞)
$Let y = \sqrt{3 x^{2} - 4 x + 5}$
⇒ y² = 3x² – 4x + 5 i.e. 3x² – 4x + (5 – y²) = 0
For x to be real, 16 – 12 (5 – y²) ≥ 0 $\Rightarrow y \geq \sqrt{\frac{11}{3}}$
$∴ Range of y = [\sqrt{\frac{11}{3}}, \infty)$

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