Question

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

• A. $R$
• B. $[4,16]$
• C. $[1,4]$
• D. $[-1,1]$

Answer 1

The correct option is C $[1,4]$

$f\left(x\right)=\sqrt{{\log }_{10}\left(\frac{5x-x^2}{4}\right)}$
$f$ is defined if
${\log }_{10}\left(\frac{5x-x^2}{4}\right)\ge 0\text{and}\frac{5x-x^2}{4}>0$
$\Rightarrow \frac{5x-x^2}{4}\ge 1\text{and}\frac{5x-x^2}{4}>0$
$\Rightarrow \frac{5x-x^2}{4}\ge 1$
$\Rightarrow 5x-x^2\ge 4$
$\Rightarrow x^2-5x+4\le 0$
$\Rightarrow (x-1)(x-4)\le 0$
$\Rightarrow x\in [1,4]$

Hence, domain of $f$ is $[1,4]$