Question

Find the domain of the following: $y = \sqrt{ [x]^2 -3[x] - 4}$ Please add the necessary calculations.

Answer 1

Answer:

For 'y' to be defined, the quantity inside the square root has to be equal to or greater than zero.

Therefore we get:

→ [x]² - 3[x] - 4 ≥ 0

→ [x]² - 4[x] + [x] - 4 ≥ 0

→ [x] ( [x] - 4 ) + 1 ( [x] - 4 ) ≥ 0

→ ( [x] + 1 ) ( [x] - 4 ) ≥ 0

⇒ [x] ∈ (-∞, -1 ] U [ 4, ∞ )

Hence we get two conditions. These are:

→ [x] ≤ -1   &   [x] ≥ 4

For the first condition, we get:

→ [x] ≤ -1  ⇔ x < 0

For the second condition we get:

→ [x] ≥ 4  ⇔ x ≥ 4

Therefore the domain of 'x' is given as:

⇒ x ∈ ( -∞, 0) U [4, ∞)

Answer 2

Step-by-step explanation:

-

+ [x] ( [X] - 4 ) + 1 ( [X] - 4) 2

+ ([x] + 1) ([x] -4)20

[x] € (-0, -1 ] [ 4,0)

Hence we get two conditions. These are: + [x] < -1 & [x] 2 4

For the first condition, we get:

- [x]<-1 + x <0

For the second condition we get:

[x] 24

Therefore the domain of 'x' is given as:

xé( -3, OU [4, o)