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When does an inverse exist?

When does an inverse exist? · Diagrams

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Tangent and arctangent exchange asymptotes

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Tangent on (−π/2,π/2) has vertical asymptotes x=±π/2; arctangent on all real inputs has horizontal asymptotes y=±π/2. Equal-scale graphs reflect in y=x; both pass through the origin.
Tangent on (−π/2,π/2) has vertical asymptotes x=±π/2; arctangent on all real inputs has horizontal asymptotes y=±π/2. Equal-scale graphs reflect in y=x; both pass through the origin.
Tangent on (−π/2,π/2) has vertical asymptotes x=±π/2; arctangent on all real inputs has horizontal asymptotes y=±π/2. Equal-scale graphs reflect in y=x; both pass through the origin.
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