If the domain of the function f(x)=[x]/(1+x²), where [x] is greatest integer ≤ x, is [2,6), then its range is

Solution

Maths sir · 3 steps

Split [2,6) at each integer, find the range on each piece, then check that the pieces overlap.

1. Freeze the integer part. On [n, n+1) we have [x] = n, so f(x) = n/(1+x²). For x > 0 this is decreasing. It takes the value n/(1+n²) at x = n and only approaches n/(1+(n+1)²), because x = n+1 is not included.

   Piece	f(x)	Range on the piece	Approximate
   [2,3)	2/(1+x²)	(1/5, 2/5]	(0.2, 0.4]
   [3,4)	3/(1+x²)	(3/17, 3/10]	(0.176, 0.3]
   [4,5)	4/(1+x²)	(2/13, 4/17]	(0.154, 0.235]
   [5,6)	5/(1+x²)	(5/37, 5/26]	(0.135, 0.192]

2. Check for gaps. Each piece's top value is larger than the open lower end of the piece before it: 3/10 > 1/5, 4/17 > 3/17 and 5/26 > 2/13. The intervals overlap, so their union has no holes.
3. So. The range is one unbroken interval from the lowest open end 5/37 to the highest value 2/5. That is (5/37, 2/5], option (b).

The idea to keep. With [x] inside a function, fix the integer on each unit interval, find each piece's range, and only then take the union. Overlapping pieces close up any gaps.

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