Transition Between Stereographic Projections

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Let \(f_{+}: S^2 - \{N\} \to \mathbb{R}^2\) denote stereographic projection from the north pole of \(S^2\), as defined in the article "Homeomorphism." Similarly, let \(f_{-} : S^2 - \{S\} \to \mathbb{R}^2\) denote stereographic projection from the south pole of \(S^2\), i.e. from the point \(S = (0,0,-1)\).

On \(\mathbb{R}^2\), the function \(g:= f_{+} \circ f_{-}^{-1} \) is well-defined. If \(g(3,4) = (a,b)\), what is \(a+b\)?

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