There are several things to be careful of here both in real problems in
science and in abstract problems in mathematics. In science the usual
warnings about units hold as in the previous example. It may *look*
OK to say where and are both in meters and then to form
, but the latter has a power series expansion and is
dimensional nonsense. In mathematics one *has* to worry about the
*domain and range* (defined below when we talk about functions).
Suppose I have the relation where is a real
dimensionless expression, and I wish to take the of both
sides. Well, the *range* of cosine is only to , and my
function is clearly strictly larger than 2.