Find the domain of the function f,defined by f(x)=√x^(3) (9-x)?

Question

$f (x) = {x^{3} (9 - x)}^{\frac{1}{2}}$ $f(x)={x^(3) (9-x)}^(1/2)$

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Answer 1 · 2018-04-15T04:37:17

This is the Domain of the function :-

$D_{f} :$ $D_f :$ $0 \leq x \leq 9$ $0<=x<=9$

NOTE that the term inside the square root MUST be Positive so these are the two necessary conditions for the function to be defined :-

$1)$ $1)$ First :-

$9 - x \geq 0$ $9-x>=0$

$\Rightarrow x \leq 9$ $rArr x<=9$

$2)$ $2)$ Second :-

$x^{3} \geq 0$ $x^3 >=0$

$\Rightarrow x \geq 0$ $rArr x>=0$

Now taking the intersection $\cap$ $∩$ of the above two conditions we get the domain of the function :-

$:. D_f :$ $0<=x<=9$

The graph of this function is shown below :-

![http://www.wolframalpha.com/input/?i=plot+%5B(x%5E3)*(9-x)%5D%5E(1%2F2)]( useruploads.socratic.org )