The domain of the function f(x)=1/√([x]²-3[x]-10) is : ( where [x] denotes the greatest integer less than or equal to x )

Solution

Maths sir · 4 steps

Solve the inequality for [x] first, then turn the integer answer into an interval for x.

1. Condition. The square root sits in the denominator, so what is inside must be strictly positive: [x]² - 3[x] - 10 > 0.

   ([x]-5)([x]+2) > 0 ⇒ [x] < -2 or [x] > 5

2. Use integer values. [x] is always an integer. So [x] < -2 means [x] ≤ -3, and [x] > 5 means [x] ≥ 6.
3. Back to x. [x] ≤ -3 holds for every x < -2, because any x in [-3, -2) still has [x] = -3. [x] ≥ 6 holds exactly when x ≥ 6, since x in (5, 6) has [x] = 5, which makes the expression zero.
4. So. The domain is (-∞, -2) ∪ [6, ∞), which is option (a).

The idea to keep. With [x] in an inequality, first solve for [x] and round to the allowed integers. Then remember [x] = n means n ≤ x < n+1, which closes the interval at the left end and opens it at the right end.

Answer from the NTA final answer key. Solution by PrepWiser.