What is a function?

As a first step in this module, we need to clearly understand the concept of a function and be able to utilize notation related to functions. Functions are used in many branches of mathematics, including calculus. We use functions to build mathematical models, often to represent the relationship between two or more variables.

For example, here is a function that relates the area of a circle to the radius of that circle: $LaTeX: A\left(r\right)\:=\:\pi\cdot r^2$ . For a given input value for r, say 10 cm, the corresponding output of this function is the area of a circle with radius 10 cm, which would be $LaTeX: A\left(10\right)\:=\:100\pi\:\:cm^2$ .

Here is one way to define a function: A function, f, is a rule or correspondence that corresponds to each element, x, in one set, called the domain of the function, exactly one element, f(x), in a second set, called the range of the function. Let's look at some other ways of expressing this definition and some examples of functions. Please watch the videos below that will illustrate the concept of a function.

We also want to recognize the difference between a relation and a function. Note that all functions are relations, but not all relations are functions. Watch the three videos below to see some different ways of illustrating the concept of a function.

Additional viewing option: Turn on closed captions with the "CC" button.

Algebra Basics: What Are Functions? is a YouTube video available at the Math Antics website.

Additional viewing option: Turn on closed captions by selecting the "CC" button and then choosing "English."

Introduction to Functions is a YouTube video available at FuseSchool - Global Education,

Additional viewing option: Turn on closed captions by selecting the "CC" button and then choosing "English."

What is a Function? is a YouTube video available at study.com.