Hi, and welcome to this overview of functions!
In this video, we will be examining the nature of mathematical relationships between variables.
As we work through some examples, our goal is to determine whether or not the criteria are met to define a special relationship known as a function.
Let’s take some time to review the basics of math rules, or relations.
When we are working with an equation, the value of variable x helps determine the value of the variable y.
y equals 2x, plus three
The x-value can be thought of as the input into the math rule that is determined by the equation, and the y-values that result are the output.
The technical terms for all x-values and y-values are "domain" and "range", respectively.
So in this equation, the range, y, is going to be the same as twice the domain, x, plus 3.
We can use a table to lay out some of the different outcomes of the equation.
So we have our equations here, and the different values of that x could be listed here.
For instance, if we find out that x is negative 2, our problem would look like negative 2 times 2 (which is negative 4), plus 3, which equals negative 1.
A set of ordered pairs can also be used to show a mathematical relationship.
The five elements of the domain and range from the table are shown here:
In words, each x-value is shown as an ordered pair with its corresponding y-value.
These ordered pairs can be graphed on a coordinate plane to show the relationship visually.
The relation, y equals 2x, plus 3, is graphed here.
The domain values are found on the x-axis, and the graph takes the shape according to the range values on the y-axis.
Another convenient visual representation of a relationship is called a mapping,
which shows a direct correspondence between the domain and range elements.
Here are two examples.
In Map A, each input value maps to one output value, but in Map B, the domain value “2” maps to two output values, 20 and 40.
This simple image plainly shows the criteria to determine whether the relation given is a function.
By definition, a relation is defined as a function if each element of the domain maps to one, and only one, element of the range.
Not all relations are functions, but functions are a subset of relations.
We have covered several representations of relations in this video.
The following image shows four representations that show a relation that does not qualify as a function.
At this point, we will focus on the graph of this relation to introduce the vertical line test,
which allows you to quickly determine the function status of any graph that you are given.
In short, if you draw a vertical line through a graph and hit only one point, then the graph represents a function.
In the example we just showed, if you draw a vertical line through the domain value of negative 2, the line will pass through the range values of 2 and negative 2.
Therefore, the relation fails the vertical line test.
Here are more examples of this convenient tool to determine a function from a graph.
Think you’ve got the hang of it?
Let’s go through a few problems together and see.
Is this a function?
(Negative 2,2), (negative 3,3), and (negative 4,4).
In this set of ordered pairs, the domain values are {negative 2, negative 3, negative 4} and the range values are {2, 3, 4}.
Because each domain value is paired with only one range value, this relation is a function.
What about this set?
(3,2), (3,3), and (2,4).
In this set, the domain values are {3, 2}, and the range values are {2, 3, 4}.
Because the domain value of 3 is paired with 2 and 3 in the range, this relation is not a function.
Here’s one more set to figure out:
(1,2), (2,2), (3,2), (4,2), and (5,2).
What do you think?
Is this a function or just a relation?
The domain of this relation is {1, 2, 3, 4, 5} and the range is {2}.
Because each element of the domain is paired with 2, it satisfies the definition of a function.
It is allowable for each element of the domain to be paired with the same element of the range.
Remember the vertical line test?
Which one of these graphs passes that test if we were to draw a vertical line through it?
The first graph here passes the test, because the vertical line only passes through one point.
In the second graph, the vertical line passes through two points, so it is not a function.
Now that we’ve gone through several different representations of functions,
you should be well on your way to knocking out more complex equations down the road.
Thanks for watching, and happy studying!