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When does an inverse exist?

When does an inverse exist? · Diagrams

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Restrict sine before taking its inverse

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Full sine has repeated outputs shown by a horizontal line. Restrict to[−π/2,π/2] with closed endpoints (±π/2,±1), then reflect to arcsine on[−1,1], range[−π/2,π/2].
Full sine has repeated outputs shown by a horizontal line. Restrict to[−π/2,π/2] with closed endpoints (±π/2,±1), then reflect to arcsine on[−1,1], range[−π/2,π/2].
Full sine has repeated outputs shown by a horizontal line. Restrict to[−π/2,π/2] with closed endpoints (±π/2,±1), then reflect to arcsine on[−1,1], range[−π/2,π/2].
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