For an example,
let f:R→R,be defined byf(x)=2x when x is rational and f(x)=3x when x is irrational.
Can it simply be concluded that the inverse is y2 when x is rational and y3 when x is irrational? Does this imply that the function is surjective?
Answer
Your conclusion is correct, but it depends on the fact that f(x) is rational when x is rational, and irrational when x is irrational.
It would be a completely different story if you had, for example,
f:R→R ,f(x)={2xif x is rational√3xif x is irrational.
In fact, in this case f would have no inverse because it is not one-to-one: f(32)=3=f(√3).
Exercise: show that this f is not onto either. Answer (roll over to reveal):
for example, there is no x such that f(x)=√3.
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