MLRF - Functions Relation and Representations Lesson

Functions Relation and Representations 

A "relation" is just a relationship between sets of information. Think of all the people in one of your classes, and think of their heights. The pairing of names and heights is a relation. In relations and functions, the pairs of names and heights are "ordered", which means one comes first and the other comes second. To put it another way, we could set up this pairing so that either you give me a name, and then I give you that person's height, or else you give me a height, and I give you the names of all the people who are that tall. The set of all the starting points is called "the domain" and the set of all the ending points is called "the range." The domain is what you start with; the range is what you end up with. The domain is the x 's; the range is the y 's.

In math, we typically think of relations with sets of numbers, specifically ordered pairs. Here is an example of a relation:

{(2,1), (3,4), (9,8)}

You can write your ordered pairs in a list, as above, and you can also present them in a table as follows:

X	Y
2	1
3	4
9	8

Mapping is another way to show a relation. A mapping looks like the following.

 

A function can also be represented by an equation such as f(x) = x. The f(x) is read as "f of x" and represents y. You can rewrite f(x) = x as y = x.

Finally, you can always use words to describe your function. For f(x) = x + 1, you might say the function rule is "a number increased by one," or "the sum of a number and one," or any other representation of adding one to a number.

We've looked at several ways in which to represent functions. Here they are listed in a table to help you remember and see the differences. These representations are all for the same function, simply represented in different ways.

 

Form	Representation
Words - verbal expression	"add one to each number"
Equation	y= x +1
Table	x 0 1 2 3 Y 1 2 3 4
Graph

A function is a special relation in which each element in the domain is mapped to only one element in the range. Let's go back to the original example of classmates and their heights. Whether or not that relation is a function depends on what our starting point is. If we are mapping heights to names, then it is not a function because more than one person may be 5'3". However, if we are mapping name to height, then the relation is a function because each person can only be one height. Therefore, the order, or mapping of the relation is important in determining whether or not a function exists.

If given a set of ordered pairs, the x values are always representative of the domain, or the starting point. We refer to the x values as the independent variable. Similarly, the y values, or dependent variable, are the range. To be a function, the domain can only map to one element in the range.

Let's watch THIS video Links to an external site. to pull it all together and then we'll work some examples.

Let us look at some relations and determine if they are functions.

 

You also saw in the above video another way to determine if a graph is a function. The method used is called the Vertical Line Test. The Vertical Line Test is a method that uses a vertical line moving across the graph to help you see if the domain is mapped to exactly one output. If the vertical line touches more than two points, then the x coordinate (domain) is matched with more than one output, and is, therefore, not a function. On the other hand, if in no place the vertical line touches more than one point, then that x coordinate is only matched to one point of the range and that relation is a function.

A relation consists of a set of ordered pairs (x, y); a relation can also be called a mapping. The x-values are the domain and the y-values are the range.

In the relation above, the domain is {1, 3, 5, 9} and the range is {2, 7, 8}. The range is only the values that have been "used" by the domain. We say the domain is the independent variables and are usually the x-values (what you input) and the range is the dependent variables and are usually the y-values (the output).

A relation can be mapped onto a graph by plotting each of the ordered pairs. This graph to the right maps the relation above.

A FUNCTION IS A RELATION IN WHICH EACH X - VALUE MAPS TO EXACTLY ONE Y - VALUE

In order for a relation to be a function, each x-value can only be associated with one y-value. It is OK if multiple x-values map to the same y-values!

Function	Not a Function
{(3, 2) (5, 2) (1, 4) (6,3)}	{(3, 2) (3, 5) (1, 7) (6, 6)}

Is it a function or not a function? Practice

1. {(3, -1) (5, -1) (7, -1) (9, -1)}
2. {(2, 3) (3, -1) (5, 6) (-2, 4)}
3. {(9, 1) (-8, 2) (3, 6) (10, 1)}
4. {(3, 5) (3, 2) (3, 1) (3, 7)}
5. {(2, -1) (5, -1) (5, 6) (-1, 2)}
6. {(-7, 8) (8, 9) (9, -7) (-7, 10)

TO VIEW THE SOLUTIONS ONCE YOU HAVE PRACTICED, CLICK HERE. Links to an external site.

Let's look at the graphical representation of relations and functions:

Notice that in a function, none of the points are on the same vertical line, but in the relation that is NOT a function two of the points are on the same vertical line.

Watch this video to get a better idea of how the Vertical Line Test works!

We use functions to tell us about relationships between values. For instance, let's say your cell phone plan charges you $0.15 per MB of data used. So we can write a function for the cost (C) in terms of the amount of data used. We would say C(m) = 0.15m.

Use the rule to complete the table for the given domain values. Write the result as an ordered pair.

Independent Variable: m MB's Used	Dependent Variable: C(m) = 0.15m	(m, C(m))
1	C(1) = 0.15(1) = 0.15	(1, 0.15)
10	C(1) = 0.15(10) = 1.50	(10, 1.50)
25	C(1) = 0.15(25) = 3.75	(25, 3.75)
100	C(1) = 0.15(100) = 15	(100, 15)
180	C(1) = 0.15(180) = 27	(180, 27)

A common misconception might be to think the domain for this function is {1, 10, 25, 100, 180} however, it is not! We know that you could use any amount of MB's of data. So we must account for those continuous values, not just the values we put in the table. The domain for this function would be: m > 0.

We know the amount of data used must be greater than 0, because you can't use a negative amount of data. But after that there are no restrictions on what the input could be!

Watch this video for a few more examples of how functions work:

Input Output Practice

1. You bought a plant that is 5 inches tall and you know that the plant will grow at a rate of 2 inches per week. Write a function for the height, h, of the plant after a certain number of weeks, w.
2. You are selling brownies at the bake sale for $0.75 each. Write a function for the revenue, R, you've earned based on the number of brownies you've sold, b.
3. Evaluate the function f(x ) = 7x - 3 for each value below: f(3), f(-2), f(0), f(-5)

TO VIEW THE SOLUTIONS ONCE YOU HAVE PRACTICED, CLICK HERE. Links to an external site.

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