what is the domain of sqrt(x(x-19)) in interval notation? ie) (-inf,2.76]?

Question

& its not (-inf,0) or (-inf,0]

Answer 1

we need: x(x - 19) ≥ 0

it will happen in two ways:
FIRST CASE:
x ≥ 0 and x - 19 ≥ 0
x ≥ 0 and x ≥ 19
Combining we get:
x ≥ 19

SECOND CASE:

x ≤ 0 and x - 19 ≤ 0
x ≤ 0 and x ≤ 19
Combining we get:
x ≤ 0

So the domain is actually:

x ∈ (-∞, 0] ∪ [19, ∞)

Answer 2

You need:

x(x - 19) ≥ 0

This happens two possible ways:

x ≥ 0 and x - 19 ≥ 0
x ≥ 0 and x ≥ 19
Combined you get:
x ≥ 19
(i.e. if x has to be greater than or equal to both 0 and 19, then it must be greater than or equal to 19)

Or

x ≤ 0 and x - 19 ≤ 0
x ≤ 0 and x ≤ 19
Combined you get:
x ≤ 0
(i.e. if x has to be less than or equal to both 0 and 19, then it must be less than or equal to 0.)

So the domain is actually:

x ∈ (-∞, 0] ∪ [19, ∞)

It's ok to take the square root of 0, so you should use ] and [ instead of ) and (.

It's easy to check by picking values:

x = -1: √((-1)(-1 - 19)) = √((-1)(-20)) = √200, which is ok
x = 1: √((1)(1 - 19)) = √(-18), which is not allowed
x = 20: √((20)(20 - 19)) =√20, which is ok

Answer 3

x>19

Answer 4

-inf (-inf,0] and [19,+inf)

Answer 5

x nor (x-19) can equal zero

x and (x-19) must both be negative or both be positive

(-inf,0) & (19,inf)

Answer 6

(19, inf)

Answer 7

what is the domain of sqrt(x(x-19))
[- infinity, 0] or - infinity and [19, + infinity]

Answer 8

(-inf,0]