Let ƒ : (1, 3) → R be a function defined by
f(x) = x[ x ]/(1 + x²) , where [x] denotes the greatest integer ≤ x. Then the range of ƒ is

Solution

Maths sir · 3 steps

Split the domain at x = 2, where [x] jumps, and find the range of each piece separately.

1. Split at the integer. On (1, 2) we have [x] = 1, so f(x) = x/(1+x²). On [2, 3) we have [x] = 2, so f(x) = 2x/(1+x²).

   d/dx(x/(1+x²)) = (1-x²)/(1+x²)² < 0 for x > 1

2. Both pieces decrease. So x/(1+x²) is decreasing for x > 1, and 2x/(1+x²), which is twice it, decreases too. Each piece takes all values between its two end values, so we only need the ends.

   Interval	f(x)	Value at left end	Value at right end	Range of piece
   (1, 2)	x/(1+x²)	1/2, not reached (x = 1 excluded)	2/5, not reached	(2/5, 1/2)
   [2, 3)	2x/(1+x²)	4/5, reached at x = 2	6/10 = 3/5, not reached	(3/5, 4/5]

3. So. The range is (2/5, 1/2) ∪ (3/5, 4/5], with a gap between 1/2 and 3/5. That is option (a).

The idea to keep. When [x] sits inside a function, break the domain at every integer. Find each piece's range on its own, check which endpoints are actually reached, and never assume the pieces join up.

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