**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 **input** to the function and y the **output**, or by calling x the **independent variable** and y the **dependent variable**.

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**Examples**
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