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When does an inverse exist?

When does an inverse exist? · Diagrams

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Cosine and its principal inverse

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Cosine restricted to[0,π], endpoints (0,1),(π,−1), reflects in y=x to arccos on[−1,1], endpoints (−1,π),(1,0), y-interceptπ/2. All endpoints included.
Cosine restricted to[0,π], endpoints (0,1),(π,−1), reflects in y=x to arccos on[−1,1], endpoints (−1,π),(1,0), y-interceptπ/2. All endpoints included.
Cosine restricted to[0,π], endpoints (0,1),(π,−1), reflects in y=x to arccos on[−1,1], endpoints (−1,π),(1,0), y-interceptπ/2. All endpoints included.
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