Inside Collection (Course): Precalculus with Engineering Applications

Summary: This module describes real world function concepts.

Why are functions so important that they form the heart of math from Algebra II onward?

Functions are used whenever *one variable depends on another variable.* This relationship between two variables is the most important in mathematics. It is a way of saying “If you tell me what

A few examples:

- "The area of a circle depends on its radius."
- "The amount of money Alice makes depends on the number of hours she works."
- “Max threw a ball. The height of the ball depends on how many seconds it has been in the air.”

In each case, there are two variables. Given enough information about the scenario, you could assert that *if you tell me this variable, I will tell you that one*. For instance, suppose you know that Alice makes $100 per day. Then we could make a chart like this.

If Alice works this many days... | ...she makes this many dollars |
---|---|

0 | 0 |

1 | 100 |

1½ | 150 |

8 | 800 |

If you tell me how long she has worked, I will tell you how much money she has made. Her earnings “depend on” how long she works.

The two variables are referred to as the *dependent variable *and the *independent variable*. The dependent variable is said to “depend on” or “be a function of” the independent variable. “The height of the ball is a function of the time.”

- "The number of Trojan soldiers depends on the number of Greek soldiers."
- "The time depends on the height of the ball."

The first of these two examples is by far the most common. It is simply not true. There may be a relationship between these two quantities—for instance, the *sum* of these two variables might be the total number of soldiers, and the* difference* between these two quantities might suggest whether the battle will be a fair one. But there is no *dependency* relationship—that is, no way to say “If you tell me the number of Greek soldiers, I will tell you the number of Trojan soldiers”—so this is not a function.

The second example is subtler: it confuses the *dependent* and the *independent* variables. The height depends on the time, not the other way around. More on this in the discussion of “Inverse Functions".

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