The domain of definition of the function f(x)=log10⁡3x−x22 is

The domain of definition of the function $f (x) = \sqrt{\log_{10} (\frac{3 x - x^{2}}{2})}$ is

A
(1, 2)
B
$(0, 1) \cup [2, \infty)$
C
[1, 2]
D
(0, 3)

Solution:

$3 x - x^{2} > 0 \Rightarrow x (3 - x) > 0 \Rightarrow x \in (0, 3)$

Also, $\log_{10} (\frac{3 x - x^{2}}{2}) \geq 0 \Rightarrow \frac{3 x - x^{2}}{2} \geq 1 \Rightarrow x^{2} - 3 x + 2 \leq 0$

$\Rightarrow x \geq 1$ or $x \leq 2 ∴ x \in [1, 2]$

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