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Functions

Introduction

A function is a mathematical relation between a set of inputs and outputs with the distinct characteristic that each input corresponds to exactly one y value. Most equations that we tend to plot are functions but there are a few examples such as circles which are not. In this lesson, you will learn how to test if an equation is a function, about domain and range and about sketching functions.

Determining if an Equation is a Function

In order to determine whether the graph of a particular equation is a function there is a very simple trick which is called the vertical line test. To apply this test, simply draw/sketch the graph of the equation you want to test. If the equation is a function, any vertical line cutting the graph at any point will only cut it once. Why does this work? It’s very simple to understand. In the definition of a function, it has been stated that there is only one Y value for each X value. If a vertical line was to cut the graph more than once, this would mean that for a given x coordinate, there is more than one y coordinate. Consequently, this would not be a function. Here are some examples. Try doing the vertical line test yourself!
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Composite Functions 20.1 COMPOSITE FUNCTIONS 20.2 INVERSE FUNCTIONS
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Domain & Range

Domain and Range is a very important concept that you will meet almost constantly when working with functions. Domain: refers to the possible input values (x values) Range: refers to the possible output values (y values) To state the domain and range, we use set notation. Let us take x² as an example. Looking at the graph, you will see that x² is valid for all values of x. So the domain does not have a restriction. However, since x²=y will always be 0 or positive, the y value must be equal to or greater than 0. The domain and range of x² can be expressed as follows: Domain: Range: There are several different formats of writing domain and range. You should ask your teacher which one you should use. Above is one of the acceptable methods. Domain and range can get more complicated as the functions get more complicated. Always watch out for things that limit the domain or range. For example, powers, negative square roots, divided by 0, etc. In many exam questions, domain and range may be used to limit the range of possible answers. For example, on a sine graph, you may be asked to find the sinx=0 for the values of x between 0 and 360 only. Make sure you watch out for these as otherwise you might end up wasting time or maybe even loosing points.

Sketching Functions

Sketching functions work like sketching all other equations (as all functions are equations). You can use technology such as graphing software or graphic calculators. Or you can sketch graphs by hand by considering axes intercepts. In the event that you are really unfamiliar with the shape of certain function, it is always best to draw up a table of x and y values around the origin and then plot the individual points before connecting them with a curve. Make sure to check out our shape section and our algebra section on more information on how to sketch graphs.

Learn more about functions

Now that you know the basics about functions, it is time to move on to the more complicated concepts such as composite functions and inverse functions. Also make sure to check out our library for additional lessons.
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Chapter 20.0:  Learning Outcomes Students will be introduced to functions! Students will learn how to identify functions! Students will learn about domain and range! Students will learn how to sketch functions!

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