1) Injection. Intuitively, we are proving that this function passes the "horizontal line test." I don't know if your professor prefers a certain method, but one approach is to take the derivative and show that f(x) is strictly decreasing.
2) Surjection. The image and the codomain are both the set of reals. This is evident since f(x) is defined everywhere and the limits as x approach negative and positive infinity are positive and negative infinity, respectively.