## Related questions with answers

Find the domain and range of the function. Write your answer in interval notation. $g(x)=\sqrt{16-x^{4}}$

Solution

VerifiedGiven function contains a square root. We must exclude all values of $x$ that give negative number in square root. In other words only $\textbf{positive}$ numbers (or zero) may occur in square root

Find the domain:

$\begin{align*} 16-x^4&\geq 0\\ 16&\geq x^4\\ x^4 &\leq 16\\\\ x \leq 2&\text{ and } x\geq -2 \end{align*}$

Thus, domain in interval notation is:

$\mathcal D_g =[-2,2]$

Square root function is increasing function, so to find its range we have to find minumum and maximum value of argument.

Minimum value is 0 when $x=-2$ or $x=2$ and maximum value is $16$ when $x=0$. Both these points are $\textbf{included}$ in the range.

Thus, range of function is $[\sqrt 0,\sqrt{16}]=[0,4]$.

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