Question

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

• A. $\left[1,4\right]$
• B. $\left[1,0\right]$
• C. $\left[0,5\right]$
• D. $\left[5,0\right]$

Answer 1

The correct option is A

$\left[1,4\right]$

Explanation of correct answer :

Finding domain of the function :

Given, $f\left(x\right)=\sqrt{{\log }_{10}\frac{5x-x^2}{4}}$

As we know, the term under root should always be non-negative, i.e.

$$\begin{gathered} {\log }_{10}\frac{\left(5x-x^2\right)}{4}\ge 0 \\ \Rightarrow \frac{\left(5x-x^2\right)}{4}\ge 1\left(\text{if}{\log }_{a}b,\text{then}b=a^c\text{and}{10}^0=1\right) \\ \Rightarrow 5x-x^2\ge 4 \\ \Rightarrow x^2-5x+4\le 0 \\ \Rightarrow x^2-4x-x+4\le 0 \\ \Rightarrow (x-4)(x-1)\le 0 \\ \Rightarrow x\in \left[1,4\right] \end{gathered}$$

As we know, for ${\log }_{a}b$ , $b$must be always greater than 0, i.e. $b>0$.

Then, $\frac{5x-x^2}{4}>0$

$$\begin{gathered} \Rightarrow 5x-x^2>0 \\ \Rightarrow x\left(5-x\right)>0 \\ \Rightarrow x\in \left(0,5\right) \end{gathered}$$

We got two values of $x$, to get the final domain, we will take intersection of both values.

Thus, the domain of the function is $\left[1,4\right]$

Hence, the correct answer is Option(A).