@John #17 – that looks basically right to me, just three small typos:

1. "The inverse image is also called the preimage, and it's often written as \\(f^{-1}(X)\\)" – I think this should be \\(f^{-1}(S)\\)
2. "The image is often written as \\(f(X)\\)" – I think this should be \\(f(S)\\)
3. "\\(\\{y \in Y:\textrm{ for some } x \in T \\}\\)" – there's a missing \\(y = f(x)\\)