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Precalculus: An Investigation of Functions, by David Lippman and Melonie Rasmussen, licensed under CC BY-SA 4.0. Read the original free This shelf’s study material is shared under the same license (CC BY-SA 4.0).
Chapter 3 · pp. 7–7
This section covers the one idea at the center of precalculus: the function. You will learn what a function is, how to evaluate one, and how to tell when a relation is not a function. Everything else in precalculus, from graphs to trigonometry, builds on this idea.
A function is a rule that gives each input exactly one output. The set of allowed inputs is called the domain. The key phrase is "exactly one." An input may never lead to two different outputs. If it does, the rule is not a function.
Think of a vending machine. If you press the button for B4, you always get the same snack. A machine that gave you chips one time and a candy bar the next, with the same button, would be unpredictable. Functions are predictable in this way.
Note that two different inputs are allowed to share the same output. For example, with , both and . This is fine. The rule only forbids one input giving more than one output.
Function notation is read "f of x." It means "the output of the rule when the input is ." To evaluate, replace with the given input and simplify.
Example: Let . This rule says "double the input, then add 3."
Each input gives one and only one output, so is a function. You can check this by noticing that the arithmetic has just one possible answer for any number you put in.
A relation is any pairing of inputs with outputs. Every function is a relation, but not every relation is a function.
Consider the circle
Here, is the input and is the output. Try :
One input, , gives two outputs, and . So this relation fails the test and is not a function of .
The same thing happens for most other -values between and . A single failure is enough, though. To show a relation is not a function, you only need to find one input with two outputs.
When you have a graph, draw imaginary vertical lines. If any vertical line touches the graph at more than one point, the graph is not a function. The circle above fails because the line crosses it at both and . The line passes the test, because every vertical line crosses it exactly once.
Functions let us describe how one quantity depends on another in a reliable way. If you know the rule and the input, you can always find the one correct output. Later topics, such as polynomial, exponential, and trigonometric functions, are all special kinds of functions with different rules.
