A map f: A - [-1, 1] defined by f(x) = sin x is required to be both one-one and onto, where A is a subset of R. Which interval A makes f a bijection?

Question

A map f: A -> [-1, 1] defined by f(x) = sin x is required to be both one-one and onto, where A is a subset of R. Which interval A makes f a bijection?

Accepted Answer

[-pi/2, pi/2]. For f(x) = sin x to be one-one, sin must be monotonic on A; for it to be onto [-1, 1], sin must attain all values from -1 to 1. The interval [-pi/2, pi/2] is exactly where sin is strictly increasing from -1 (at -pi/2) to 1 (at pi/2), covering all of [-1, 1] once. The other intervals either repeat values (not one-one) or don't span the full range. Hence A = [-pi/2, pi/2].

A map f: A -> [-1, 1] defined by f(x) = sin x is required to be both one-one and onto, where A is a subset of R. Which interval A makes f a bijection?

Solution

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