proving equal cardinality -- proof writing examples 22

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[M.athematech]

for the first example for surjectivity it should be explicitly noted that 1/b - 1 is positive so that it is in the domain of the natural log function, and for he second example it should be shown that a so defined is an integer

[Happy_Abe]

When b<0, it should be f(1, |b|+1)

[Grizzly]

Please explain why g(x) = 1/(1 + e^x) was chosen as the first function.
Am I correct in assuming that any function of that form would suffice, eg. 1/(1 + 2^x), as long as the denominator is ≥ 1?