Find the domain and range of the following functions. f(x)=16-x^2 - Vidyadip

Question

Find the domain and range of the following functions.
$f(x)=\sqrt{16-x^2}$

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Answer

$f(x)=\sqrt{16-x^2}$
For $f$ to be defined,
$ 16-\mathrm{x}^2 \geq 0$
$\therefore \mathrm{x}^2 \leq 16$
$\therefore-4 \leq \mathrm{x} \leq 4 $
$\therefore$ Domain of $f=[-4,4]$
Clearly, $f(x) \geq 0$ and the value of $f(x)$ would be maximum when the quantity subtracted from $16$ is minimum i.e. $x=0$
$\therefore$ Maximum value of $f(x)=\sqrt{ 16}=4$
$\therefore$ Range of $f=[0,4]$

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