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  4. The domain of the function f(x)=√ log 10((5 x-x2/4)) is

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Q. The domain of the function $f(x)=\sqrt{\log _{10}\left(\frac{5 x-x^{2}}{4}\right)}$ is

TS EAMCET 2020

Solution:

We have
$f(x)=\sqrt{\log _{10}\left(\frac{5 x-x^{2}}{4}\right)}$
For $f(x)$ to be defined
$\Rightarrow \frac{5 x-x^{2}}{4} \geq 1 $and $ \frac{5 x-x^{2}}{4}>0 $
$\Rightarrow \frac{5 x-x^{2}}{4} \geq 1 $
$\Rightarrow 5 x-x^{2} \geq 4$
$ \Rightarrow x^{2}-5 x+4 \leq 0 $
$\Rightarrow (x-4)(x-1) \leq 0$
image
$\therefore x \in[1,4]$
So, domain of $f(x)$ is $[1,4\}$