# What is the domain of the function f(x) =sqrt((x-1)/(x+4))

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`sqrt((x-1)/(x+4))`

`` The domain is all real value where the function is defined.

The square root is not defined for negative values.

`==gt (x-1)/(x+4) gt 0`

`` Also, the denominator should not be zero.

`==gt x!= -4 `

`==gt (x-1)gt 0 and (x+4)gt 0 `

`==gt x gt= 1 and x gt -4`

`` `==gt x in [1,oo).`

`` Also,`(x-1) lt= 0 and (x+4) lt 0`

`` `==gt x lt= 1 and x lt -4`

`` `==gt x in (-oo, -4)`

`==> x in (-oo,-4)U[1,oo).`

``

``

The value under the square root should be positive so:

x-1>0 and x+4>0 or x-1<0 and x+4<0 ==>

x>1 and x>-4 or x<1 and x<-4 ==>

x>1 or x<-4

so **Domain=(-inf,-4)U(1,inf)**