I am confused about how to represent a function to fix one on one correspondence, and how to represent a function only correspondence but not onto. Here are the example questions, could someone give me help or a tip?
Find a function f : [1, 2) → (1, 2) that is 1-1 correspondence.
f: (0.5, 1) → (0, 1) that is 1-1 but not onto;
f: (0.5, 1) → (0, 1) that is 1-1 correspondence;
On
Your question needs some clarification: do you need to find an $f$ that is 1-1, i.e. injective, or do you need to find a 1-1 correspondence between the intervals, i.e. a bijection?
The problem is pretty easy if it's the first, so let's assume the second.
The obvious difficulty is the $1$ in the interval $[1,2)$, so think about where it will map to. Since we must have $1 < f(1) < 2$, $f(1)$ partitions the range into two open intervals $(1, f(1))\cup(f(1),2)$. Now the problem is to find a bijection between $(1,2)$ and those two open intervals.
An answer is already provided by William Elliot.
Put $f(x)=x$ for each $x$ from $(0.5,1)$.
Put $f(x)=2x-1$ for each $x$ from $(0.5,1)$.