Represent a mapping between two sets as a list of pairs--one for each element in the set. For example, if element A in one set is linked to an element B in the other set, we write (A, B). If the first element is plotted on the X axis and the second element is plotted on the Y axis, the test to see if the map is a function is called the "vertical line test." If no vertical line crosses the curve more than once, the curve is a "function."
Define an "injection" as a map where each element in the first set goes to a unique element in the second set. Not all functions are injective. In the list of pairs, every element of the first set must appear as the first element of a pair, and all of the second elements of the pairs must be different. The test to see if a function is injective is the "horizontal line test." If no horizontal line cuts through the curve in more than one place, the function is injective.
Use the inverse mapping theorem to find the inverse of a map. The theorem says that if an inverse exists, it can be found in three steps: 1) write the map in graph notation; 2) exchange the variables; 3) the solution is the inverse map of the original. For example, if the function is y = 3x - 1, reversing the variables gives x = 3y - 1. Solving gives y = 1/3(x + 1) which is the reverse map. Not every function has an inverse. An example is y = x^2 , because both 2 and -2 map to 4, so the inverse would map 4 to two different elements.
Answer the question about when are maps reversible very simply: a map is reversible if, and only if, it's injective and the sets have the same cardinality.