What is a Function?
Definition. A function is a rule which relates the values of one variable quantity to the values of another variable quantity, and does so in such a way that the value of the second variable quantity is uniquely determined by (i.e. is a function of) the value of the first variable quantity.
This is clearly a very general definition. Such rules are general enough to serve as mathematical models for a large number of natural relationships between variable physical quantities.
We name functions by using symbols like "f," "g," "h," etc. Just as your name signifies all of the many things that make "you," a symbol like "f" serves as a shorthand for what may be a long or complicated rule expressing a particular relationship between variable quantities. We name the variable quantities with symbols like "x," "y," "t," etc. We then write
Notice that the definition distinguishes between a first variable quantity and a second variable quantity. It is the second variable which is a function of the first, and when we write
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