Specifically, a function is invertible if it is 1:1, that is there is one and only one y value for each x. You can use the "horizontal line test" to determine this: pass a horizontal line through the graph of the function. If the line cuts the graph of the function in more than one place then the function has no inverse.
Note that we CAN have a (piece-wise defined) function that is increasing on some intervals and decreasing on others that is invertible.
-Dan