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The fractional part of x is denoted {x}. Find the range of f(x) = {x}/(1 + {x}).

1. [0, 1)
2. [0, 1/2]
3. [0, 1/2)
4. (0, 1/2)

Correct answer: [0, 1/2)

Solution

Let t = {x} ∈ [0, 1). Then f = t/(1+t) = 1 - 1/(1+t), which increases with t. At t=0, f=0 (attained). As t→1⁻, f→1/2 but never reaches it (since t=1 is excluded). So range = [0, 1/2).

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