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

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# f(x)={x^(3) (9-x)}^(1/2)#

This is the Domain of the function :-

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

Now taking the intersection

The graph of this function is shown below :-

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To find the domain of the function ( f(x) = \sqrt{x^3 (9-x)} ), we need to consider the values of ( x ) that make the expression under the square root valid. The expression under the square root, ( x^3 (9-x) ), must be non-negative, so:

[ x^3 (9-x) \geq 0 ]

The critical points are where the expression equals zero, namely ( x = 0 ) and ( x = 9 ). We can test intervals defined by these critical points on the number line.

- For ( x < 0 ), both ( x^3 ) and ( (9-x) ) are negative, so their product is positive.
- For ( 0 < x < 9 ), ( x^3 ) is positive and ( (9-x) ) is positive, so their product is positive.
- For ( x > 9 ), ( x^3 ) is positive and ( (9-x) ) is negative, so their product is negative.

Therefore, the domain of ( f(x) ) is ( {x \in \mathbb{R} : x \leq 0 \text{ or } 0 \leq x \leq 9} ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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