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Find the domain of the real-valued function

\[f(x)=\sqrt{-6x^2+12x+18}.\]

Give the endpoints in your answer as common fractions (not mixed numbers or decimals).

 Aug 19, 2021
 #1
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The stuff under square root needs to be bigger than 0.

 

-6x^2+12x+18>= 0

 

We factor to get: -6(x^2-2x-3)>= 0, thus -6(x-3)(x+1) >= 0.

 

We can get rid of -6, but we need to change the sign. (x-3)(x+1) <= 0.

 

Two pairs of possibilities: x-3>= 0, x+1 <=0

and x-3 <= 0, x+1 >=0.

 

Simplify to get x>=3, x<=-1(which is impossible so we abandon it)

and x<=3, x>=-1(which makes sense)

 

Finally, the domain is [-1, 3]

 Aug 19, 2021

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