Find the domain of the following function. State your answer in interval notation. f(x)=(8x-12)/(x+sqr(-8x-16))   please help

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`f(x)= (8x-12)/(x+sqrt(-8x-16))`

`==> x+sqrt(-8x-16) !=0 and -8x-16 >=0`

`==> x+ sqrt(-8x-16) != 0`

`==> x != -sqrt(-8x-16)`

`==> x^2 != -8x-16`

`==> x^2 +8x+16 != 0`

`==. (x+4)^2 !=0`

`==> x != -4`

`==> x in (-oo,-4)U(-4,00)`

``

`-8x-16 >= 0`

`==> -8x >= 16`

`==> x <= -2`

`==> x in (-oo, -2]`

Then, the domain must satisfies both (1) and (2).

==> `x in (-oo, -2] nn (-oo,-4) U (-4,oo)`

`==> x in (-oo,-2]nn(-oo,-4) U (-oo, -2] nn (-4,oo)`

`==> x in (-oo, -4) U (-4, -2]`

 

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