**Introduction**

Calculus is the study of change. It is often important to know when something is increasing, when it is decreasing, and when it hits a high or low point. Much of the business of finance depends on predicting the high and low points for prices. In science and engineering, it is often essential to know precisely how fast quantities such as temperature, size, and speed are changing. Calculus is the primary tool for calculating such changes.

Numbers, which are the focus of arithmetic, are no longer the objects of our study. This is because they do not change. The number 5 will always be 5. It never goes up or down. Thus, we need to introduce a new sort of mathematical object, something that *can* change. These objects, the centerpiece of calculus, are functions.

**Functions**

A *function* is a way of matching up one set of numbers with another. The first set of numbers is called the *domain*. For each of these numbers in a set, the function assigns exactly one number from the other set, the *range*.

For example, the domain of the function could be the numbers 1, 4, 9, 25, and 100; and the range could be 1,2,3,5, and 10. Suppose the function takes 1 to 1,4 to 2, 9 to 3, 25 to 5, and 100 to 10. This could be illustrated by the following:

1 → 1 |

4 → 2 |

9 → 3 |

25 → 5 |

100 → 10 |

Because we sometimes use several functions at the same time, we give them names. Let us call the function we just mentioned by the name *Eugene*. Thus, we can ask, "Hey, what does Eugene do with the number 4?" The answer is "Eugene takes 4 to the number 2."

Mathematicians are notoriously lazy, so we try to do as little writing as possible. Thus, instead of writing "Eugene takes 4 to the number 2," we often write "Eugene(4) = 2" to mean the same thing. Similarly, we like to use names that are as short as possible, such as *f* (for function), *g* (for function when *f* is already being used), *h*, and so on. The trigonometric functions in Lesson 4 all have three-letter names like sin and cos, but even these are abbreviations. So let us save space and use *f* instead of Eugene.

Because the domain is small, it is easy to write out everything:

f(1) = 1 |

f(4) = 2 |

f(9) = 3 |

f(25) = 5 |

f(100) = 10 |

However, if the domain were large, this would get very tedious. It is much easier to find a pattern and use that pattern to describe the function. Our function *f* just happens to take each number of its domain to the square root of that number. Therefore, we can describe *f* by saying:

*f*(a number) = the square root of that number

Of course, anyone with experience in algebra knows that writing "a number" over and over is a waste of time. Why not just pick a *variable* to represent the number? Just as *f* is our favorite name for functions, little *x* is the most beloved of all variable names. Here is the way to represent our function *f* with the absolute least amount of writing necessary:

*f*(*x*) = √*x*

This tells us that putting a number into the function *f* is the same as putting it into √ . Thus,

*f*(25) = √25 = 5 and *f*(4) = √4 = 2.

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