Proving a function is a one-to-one correspondence
Determine whether each of the following functions is one-to-one, onto, neither or both.
, given by
So, I think this is one-to-one and onto. So i need to prove it.
Claim: If , given by , then f is a one-to-one correspondence.
Proof: Assume , given by .
First we must show that f is one-to-one.
Let such that .
Hence, f is one-to-one. Here, do I need to use the fact that the codomain is (1, infinity) or the domain is (2, infinity)?
Now we need to show that f is onto. Let and take a = ....
Here I would have solved b = a / (a-2) for a, but I do not know how to solve this. Is there any other way to do this proof besides solving for a then substituting back in to show that it gives me f(a) = b?