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Find the range of the function f(x) = (3 + x - [x]) / (5 + x - [x]), where [x] denotes the greatest integer function.

1. [3/5, 4/6)
2. (3/5, 2/3]
3. [3/5, 2/3)
4. [1/2, 2/3)

Correct answer: [3/5, 2/3)

Solution

Let t = {x} = x - [x], with 0 <= t < 1. Then f = (3 + t)/(5 + t) = 1 - 2/(5 + t). As t increases from 0 to just below 1, 5 + t increases from 5 to just below 6, so 2/(5+t) decreases from 2/5 to just above 2/6 = 1/3, and f = 1 - 2/(5+t) increases from 1 - 2/5 = 3/5 to just below 1 - 1/3 = 2/3. Since t = 0 is attained but t = 1 is not, f ranges over [3/5, 2/3).

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