- A SKEPTIC's GUIDE - A SKEPTIC's GUIDE 
Mathematics is often said to be the language of Physics. This is not the whole truth, but it is part of the truth; one ubiquitous characteristic of Physics (the human activity), if not physics (the supposed methodology of nature), is the expression of relationships between measurable quantities in terms of mathematical formulae. The advantages of such notation are that it is concise, precise and "elegant," and that it allows one to calculate quantitative predictions which can be compared with measured experimental results to test the validity of the description.
The nearly-universal image used in such mathematical descriptions of nature is the FUNCTION, an abstract concept symbolized in the form y(x) [read "y of x"] which formally represents mathematical shorthand for a recipé whereby a value of the "dependent variable" y can be calculated for any given value of the "independent variable" x.
The explicit expression of such a recipé is always in the form of an equation. For instance, the answer to the question, "What is y(x)?" may be " y = 2 + 5x2 - 3x3." This tells us how to get a numerical value of y to "go with" any value of x we might pick. For this reason, in Mathematics (the human activity) it is often formally convenient to think of a function as a mapping - i.e. a collection of pairs of numbers (x,y) with a concise prescription to tell us how to find the y which goes with each x. In this sense it is also easier to picture the "inverse function" x(y) which tells us how to find a value of x corresponding to a given y. [There is not always a unique answer. Consider y = x2.] On the other hand, whenever we go to use an explicit formula for y(x), it is essential to think of it as a recipé - e.g. for the example described above, "Take the quantity inside the parentheses (whatever it is) and do the following arithmetic on it: first cube whatever-it-is and multiply by 3; save that result and subtract it from the result you get when you multiply 5 by the square of whatever-it-is; finally add 2 to the difference and voila! you have the value of y that goes with x = whatever-it-is."
This is most easily understood by working through a few examples, which we will do shortly.