The domain of the definition of the function -log 105 x - x 2-4A- -1-4-B- 1-4-C- 0-5D- -0-5-

The correct option is A [1, 4]For function to be defined, nbsp; log _10 ( 5x x^24 ) ≥ 0 and 5x x^24 gt;0 ⇒ nbsp; 5x x^24≥1 and 5x x^2 gt;0 ⇒ 5x x^2 4 ...

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Standard XII

Mathematics

Domain

The domain of...

Question

The domain of the definition of the function $\sqrt{{log}_{10} (\frac{5 x - x^{2}}{4})}$ is

[1, 4]

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(1, 4]

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(0, 5)

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[0, 5]

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Solution

The correct option is A $[1, 4]$
For function to be defined,
${log}_{10} (\frac{5 x - x^{2}}{4}) \geq 0 and \frac{5 x - x^{2}}{4} > 0 \Rightarrow \frac{5 x - x^{2}}{4} \geq 1 and 5 x - x^{2} > 0$
$\Rightarrow 5 x - x^{2} - 4 \geq 0 and x^{2} - 5 x < 0 \Rightarrow x^{2} - 5 x + 4 \leq 0 and x (x - 5) < 0 \Rightarrow (x - 4) (x - 1) \leq 0 and 0 < x < 5 \Rightarrow 1 \leq x \leq 4 and 0 < x < 5 \Rightarrow x \in [1, 4]$

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The domain of the definition of the function √log10(5x−x24) is

The correct option is A [1,4]For function to be defined, log10(5x−x24)≥0 and 5x−x24>0⇒5x−x24≥1 and 5x−x2>0 ⇒5x−x2−4≥0 and x2−5x<0⇒x2−5x+4≤0 and x(x−5)<0⇒(x−4)(x−1)≤0 and 0<x<5⇒1≤x≤4 and 0<x<5⇒x∈[1,4]

The domain of the definition of the function $\sqrt{{log}_{10} (\frac{5 x - x^{2}}{4})}$ is