Penalty on Exam: Severe
Each "type" of function you encounter in mathematics has its own set of unique properties and rules. Anyone who has worked with logarithmic functions knows that there is a list of unique properties for manipulating such functions. Those who are familiar with trigonometric functions knows that there is a "laundry list" of properties associated with these functions as well.
While it should be obvious to the student that they must know all of the properties for the functions they are working with, there is a relatively easy way to avoid the mistakes listed here. Before describing the method. however, we must first think of each incorrect action in the example list above as "splitting" or "breaking" the function. The first example "split" the squaring action into two squares. The second example "split" the square root into two square roots. The third "split" the logarithm into two logarithms. The fourth "broke" the 2 out of the sine function.
Now that we have an idea of what we were doing to get those incorrect answers, let's devise a loose "rule" to avoid these mistakes in the future. It can be summarized in the following statement.
Just before you "split" or "break" a function, ask yourself if you have learned a property that explicitly tells you to perform that "split" or "break" with this type of function.
You will often find the answer to that question is a resounding "no."