Question

The domain of definition of $f(x)=\sqrt{4x-{{x}^{2}}}$ is
A. $R-[0,4]$
B. $R-(0,4)$
C. $(0,4)$
D. $[0,4]$

Answer 1

Hint: We need to find the domain of $f(x)=\sqrt{4x-{{x}^{2}}}$ . First, we will find the inequality of this function. As the square root of a number is always positive, $4x-{{x}^{2}}\ge 0$ . Find the values of $x$ for which $4x-{{x}^{2}}$ is positive. Then plot the graph. From these, the domain of $f(x)$ can be obtained.

Complete step by step answer:
We need to find the domain of $f(x)=\sqrt{4x-{{x}^{2}}}$ .
We know that the square root of a number is always positive.
i.e. $4x-{{x}^{2}}\ge 0$
Now, let $y=4x-{{x}^{2}}...(i)$
We have to plot the graph of this function.
When $x=0,$
Substituting the value of $x$ in $(i)$ , we get
$y=4\times 0-{{0}^{2}}$
$\Rightarrow y=0$
When $x=1,$
Substituting the value of $x$ in $(i)$ , we get
$y=4\times 1-{{1}^{2}}$
$\Rightarrow y=3$
When $x=2,$
Substituting the value of $x$ in $(i)$ , we get
$y=4\times 2-{{2}^{2}}$
$\Rightarrow y=4$
When $x=3,$
Substituting the value of $x$ in $(i)$ , we get
$y=4\times 3-{{3}^{2}}$
$\Rightarrow y=3$
When $x=4,$
Substituting the value of $x$ in $(i)$ , we get
$y=4\times 4-{{4}^{2}}$
$\Rightarrow y=0$
Further value of $x$ gives negative values of $y$ .
For example,
When $x=5,$
Substituting the value of $x$ in $(i)$ , we get
$y=4\times 5-{{5}^{2}}$
$\Rightarrow y=-5$
As $y=4x-{{x}^{2}}\ge 0$ , we will take the values till $x=4$ .
Let us plot the graph.

It is clear that the domain is $[0,4]$ since $4x-{{x}^{2}}\ge 0$ .
Hence the correct option is D.

Note:
Domain of a function can be easily obtained through the graph. The domain will be determined according to the inequality obtained. The domain can also be obtained without drawing the graph. From the sections where the $x$ and $y$ values are determined, those values of $x$ for which the value of $y$ is positive will be the domain.