First slide

Functions (XII)

Question

The range of the function f (x) = log_e (3x² – 4x + 5) is

Moderate

A

$(- \infty, \log_{e} \frac{11}{3}]$
B

$[\log_{e} \frac{11}{3}, \infty)$
C

$[- \log_{e} \frac{11}{3}, \log_{e} \frac{11}{3}]$
D

None of these

Solution

f (x) is defined if 3x² – 4x + 5 > 0
$\Rightarrow 3 [x^{2} - \frac{4}{3} x + \frac{5}{3}] > 0 \Rightarrow 3 [{(x - \frac{2}{3})}^{2} + \frac{11}{9}] > 0$
which is true for all real x.
$∴$ Domain ( f ) = (– ∞, ∞)
$y = \log_{e} (3 x^{2} - 4 x + 5) \Rightarrow e^{y} = 3 x^{2} - 4 x + 5 \Rightarrow 3 x^{2} - 4 x + (5 - e^{y}) = 0$
For x to be real,
$16 - 12 (5 - e^{y}) \geq 0 \Rightarrow 12 e^{y} \geq 44 \Rightarrow e^{y} \geq \frac{11}{3} \begin{array}{l} \Rightarrow y \geq \log_{e} \frac{11}{3} \\ ∴ Range of f = [\log_{e} \frac{11}{3}, \infty) \end{array}$

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