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

Question

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

James Allen · Accepted Answer

This is the Domain of the function :-

#D_f : ##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)# First :-

#9-x>=0 #

#rArr x<=9#

#2)# Second :-

#x^3 >=0#

#rArr x>=0#

Now taking the intersection #∩ # 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 :-

Serenity Jones · Answer

undefined 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.

1. For $ x < 0 $, both $ x^3 $ and $ (9-x) $ are negative, so their product is positive.
2. For $ 0 < x < 9 $, $ x^3 $ is positive and $ (9-x) $ is positive, so their product is positive.
3. 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\} $.