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Question:

Let f: R → R be defined by f(x) = x/(1+x²), x ∈ R. Then the range of f is:

(-1, 1)

[-1/2, 1/2]

R - [-1, 1]

R - [-1/2, 1/2]

Solution:

f(0) = 0. f(x) is odd. Further, if x > 0 then f(x) = x/(1+x²) ∈ (0, 1/2]. Hence, f(x) ∈ [-1/2, 1/2]