What Is a Set?
First, let's understand sets. A set is a collection of things or objects. It is usually a collection of numbers, but it can also be a collection of other things, such as letters, names, or almost anything. For example, the set of all integers greater than 0 and less than 10 is {1, 2, 3, 4, 5, 6, 7, 8, 9}.
Mapping
Now that you know what a set is. To understand functions, let's name two sets: the input set ("domain") and the output set ("range"). A function takes each value from the input set and maps it to a value in the output set. For example: y = 2x + 1. In this equation, we consider "x" as the set of numbers we can input (domain), and "y" as the set of output numbers we get (range). Then y = 2x + 1 is the function that relates the two sets of numbers.
| Input Set ("Domain") | Output Set ("Range") |
| 1 | 3 |
| 2 | 5 |
| 3 | 7 |
| 4 | 9 |
| 5 | 11 |
| 6 | 13 |
| 7 | 15 |
| 8 | 17 |
Important Properties of Functions
Functions do not allow "one-to-many" relationships, meaning that a specific input value cannot produce multiple different output values. For example, y² = x; when x = 9, y can be either 3 or -3. We would not call this a function.
How to Write a Function
Returning to the function y = 2x + 1, we usually represent y with f(x). Here, f is not a variable name, but an abbreviation for the function name. Therefore, we often write the function as f(x) = 2x + 1.
Types of Functions
You can observe that f(x) = 2x + 1 has a corresponding relationship of (x, y), {...(1,3), (2,5), (3,7)...}. And when you graph all these input and output value pairs on a coordinate plane, you will find that it is a straight line, which is called a "linear function". In algebra, there are many different types of functions, and their graphs are all interesting: quadratic functions, cubic functions, trigonometric functions, and so on.
1. Linear Function
2. Quadratic Function
3. Cubic Function
4. Trigonometric Function
Summary
In algebra, functions usually appear in the form of equations, and the graphs of these equations can be plotted on a coordinate plane by considering the input and output values as ordered pairs. If you are worried about making mistakes in graphing, you can use our free algebra calculator to quickly get the answer and graph by entering your algebra problem.
