Lesson 2 - Functions
Unit 1
Chapter 2 - Functions
LESSON 2.1 - RELATIONS AND FUNCTIONS
- A relation is a rule that connects two or more things. It can be a connection between objects, people, numbers, etc. or any combination thereof. A relation tells us what to do with input to get a specific output.
- Consider, for example, the relation defined by the rule “the sum of all the digits in a number”. If our input is 23, our output would be 2 + 3 = 5. If the input is 148, our output would be 1 + 4 + 8 = 13, and so on.
- The input is often named the independent variable, since it does not depend on any other factor, and the output is the dependent variable, since its value does depend on the input. We cannot know what the output will be until we know what the input is.
- One way to represent a relationship is with an arrow diagram. This is where all input values are placed in one circle, and all output values for a certain rule are placed in another. An arrow (or a line) then connects the input values to the corresponding output values.
Before arrows are added, it would look like this:
For the rule "a number times two", the completed diagram would look like this:
Each line connects an input value to the output it matches based on the given rule.
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A relation can be described in many ways. Some ways are more useful than others, depending on the context. Imagine a relation that connects hundreds of values to one another—an arrow diagram would not be the most effective way to display this.
In this lesson, we will explore seven different representations that are quite common. Do not worry if you do not fully understand each one yet; this lesson is an introduction to the concepts that we will study in much more detail later on.
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Functions
- The term functionis something we have become familiar with; it has been used quite a bit already in this course, but we have not yet defined what a function actually is.
- A function is a special kind of relation in which each valid input gives exactly one output.
- It is important to note that all functions are relations; however, not every relation is a function.
Take a look at the two relations below:
The first relation is a function. Each of the input values corresponds to exactly one output. Even though both -1 and 1 have the same output value, we are able to say an input of -1 connects to exactly one output of 1.
The second relation is not a function. Notice how an input of 1 corresponds to an output of both 1 and 2. The input of 1 does not correspond to exactly one output value, so this relation does not meet the definition of a function.
Example 1
Solution
Functions
When we are given a graph, we can determine if it is a function with the vertical line test. We can use this test by drawing an imaginary vertical line on a graph of the relation.
- If the vertical line intersects the graph exactly once, then it passes the vertical line test, and it is a function.
- If it intersects the graph more than once, then it fails the vertical line test, and it is not a function.
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Common Mistake
- Note that the vertical line test must be used on all possible vertical lines before deciding whether the graph is a function or not. Imagine your vertical line travelling all the way across your graph form left to right- and ask the question: "Can I find at least one place that the line crosses the graph at more than one point?" If you cannot, then the graph is a function.
Example 2
State whether each of the following pass or fail the vertical line test.
Solution
- This relation passes the vertical line test and is therefore a function.
- This relation fails the vertical line test and is therefore not a function.
- This relation passes the vertical line test and is therefore a function.
State whether each of the following pass or fail the vertical line test.
- For the following relations, identify whether or not they are a function.
a) Function
b) Non-function
c) Function
d) Non-Function
e) Function
f) Function
g) Non-Function
h) Non-Function
i) Non-Function