LE 2 Traditional Math: Relations and Functions
Accelerated seventh grade math and regular eighth grade math
The previous section defined what a function is. This section steps back from functions and looks at the more general term “relation” which is a set of ordered pairs of numbers. There is a distinction between relations that are functions and relations that are not. In general, textbooks focus on identification; e.g., providing tables, or ordered pairs and asking students to determine which are functions and which are not. The determination of a function is whether each element of the domain has one and only one element in the range. This section provides examples of what that means to allow students to more easily grasp the concept of one input/one output—a concept that often leaves them confused.
The reason for such confusion is that the importance of relations is not apparent given the limited exposure they have had to concepts like square roots and absolute value that can result in different outputs for the same input. For example if y = √x, you will have two output values for each positive x, so it is not a function. Given this situation, I don’t dwell on relations other than to get the concept across so they know what it is, and then move on.
This section also covers functional notation: f(x) and shows how it is used. Students are given practice finding function values and writing functions. The section functions as a bridge to solidify the concept of function and the various terms such as dependent and independent variables, and domain and range of a function.
Warm-Ups. These cover some of the material in the previous lesson and include older concepts.
1. Multiply. -3/5 × 5/17 Answer: -3/17
2. The function d = 10t gives the distance (d) a bicycle travels at 10 mph for a specific time (t) in hours. Use the function to find out how long it takes for the bike to travel 25 miles. Answer: 25 = 10t; t = 2.5 hours
3. For Problem 2, what is the dependent variable and the independent variable. Answer: d is the dependent variable; t is the independent variable.
4. Libby paid $600 for a watch at a discount of 20%. Scott paid $630 for the same watch. What was the percent discount given to Scott? Answer: Original amount for watch: 100-20/100 = 600/x; 0.8x = 600; x=$750. Discount for Scott: (750-630)/750 = 120/750 =0.16 = 16%
5. Fill in the y values of the table if y = |x|.
Problem 2 requires the student to find a value using the function. This may raise the question of what the difference is between an equation and a function. If this question comes up, I remind students that a function takes input values and assigns output values to them. Sometimes they can be random, and other times they can be assigned by a rule which may be an equation. Problem 5 is used to build off of in the discussion of the difference between a relation and a function.
The Wind-Up: The Definition of Function Again. I start with a review of the definition of function. “You will recall that we said ‘A function is a rule which assigns to each input value one and only one output value.’ I also said I’d get into the one and only one output value in the next lesson. Well, now’s the time.
What this means is that a function is a set of ordered pairs in which no two pairs have the same first number and different second numbers.
“So for a set of ordered pairs of numbers to be a function, the first number of each ordered pair must correspond to a unique second number.”
I now take a closer look at Problem 5 of the Warm-Ups which we have gone over. “The function there had x be any number, and y be the absolute value of that number. We saw that the resulting table had the same second number for different first numbers; specifically we had (1,1) and (-1,1). Is that a function?”
There will be some discussion in which I point them to what I said. “We can have no two pairs with the same first number and different second numbers. Is that what we have in Problem 5?” To further hammer home this point, I give an example where we can have different first numbers and the same second number for each one. I will write the following table on the board:
“Is this a function?”
Some will say yes, but some students will say no. We explore this and always refer to the definition. I do this because I want students to get used to working with definitions and theorems as our reference points in working with mathematical arguments.
“We don’t have the same first number for different second numbers, so it complies with the definition. We can in fact have all the second numbers the same for different first numbers. It’s similar to mail deliveries. You can get more than one letter delivered to your address. What can’t happen is delivering one letter to more than one address.”
This last comparison seems to get some “Ohhh, I see” reactions.
“Let’s take a look at Problem 5 again, but this time, let’s let the x values represent the absolute values of the y values.” I write the following table on the board with the y values filled in:
“Now let’s find the absolute value of each of the y values to get the various x values.”
We end up with the following:
“Is this a function?” Since there are different y values for the same x values, this does not qualify as a function.
The Pitch: What is a Relation? “We’ve seen that sometimes sets of ordered pairs can be functions, and some of them are not. Functions are a special case for a more general concept called a relation.” I have them write the following in their notebooks:
A relation is a set of ordered pairs of numbers.
“This means that some relations can be functions. If a relation has ordered pairs in which no two pairs have the same first number and different second numbers, then that relation is a function. If a relation has two or more ordered pairs of numbers in which the first numbers are the same but the second numbers are different, than that is not a function.
Examples.
Are these relations functions? Why or why not?
1. (36,6), (36,-6), (25, 5), (25, -5), (30, 5), (31, 5)
Answer: No. It has the same first numbers and different second numbers for the ordered pairs.
2. (0,0), (1,1), (2,2), (3,3), (4,4)
Answer: Yes, it has one and only one output for each input value.
3. (2/4, 2), (3/6, 5), (4/8), 6)
Answer: No; the first numbers are the same; they all reduce to ½ and have different second numbers.
Vertical Line Test. Relations can be graphed. The graph itself can be used to determine whether it represents a function or not.
I will put a graph up on the board, or project it:
“We can draw a vertical line to the graph of the relation. If for each x value in the domain, the vertical line passes through no more than one point of the graph, it represents a function.”
I’ll have someone draw a vertical line. “Put the vertical line anywhere you want.”
“Is this a function?” In fact, it is. We can see that the vertical line if placed anywhere would not result in it passing through more than one point of the graph.
I put another graph on the board:
I’ll have someone draw a vertical line as before.
“Does the line pass through more than one point?”
In fact, it does, so it is not a function. There are two different y values for the same x value.
Functional Notation. In yesterday’s lesson we talked about “rules” that assign output values to input values.
“As we saw, these are equations that have two variables, like y = 2x + 3. This is a rule that relates the x and y variables. When x equals zero, what does y equal?”
They should get “3” for an answer, and this should be familiar to them.
“We can write these rules as equations like we just did. The y represents the dependent variable. But we can use function notation, and instead of writing the letter y for the dependent variable we can write f(x).”
Some students look at this with curiosity; others with fear and horror. “It’s just a symbol and it means the same thing as y. It’s pronounced ‘the f of x’ which means ‘the value of the function at x’.”
I provide some examples of how this works. “Say we have f(x) = 2x -1. Then f(0) means the value of the function when x equals 0. So what is that value?”
Students see fairly quickly that it is a plug-in, and tell me the value is -1. Other examples then follow.
Examples.
1. If f(x) = 4x – 7 find f(6). Answer: f(6) – 7 = 17
2. If f(x) = 14 + 3x find f(4). Answer: 14 + 3(4) = 14 + 12 = 26
3. If f(x) = 6x-4 find f(-5). Answer: 6(-5) -4 = -30-4 = -34
4. Downloading songs from an online music store costs $0.90 per song. Write a function where x represents the number of songs and f(x) is the total cost. Answer: f(x) = 0.9x
5. In Problem 4, find the cost if 4 songs are downloaded. Write your work in functional notation. Answer: f(4) = 0.9(4) ; f(4) = $3.60
6. For Problem 4, if f(x) = $22.50, how many songs were downloaded? (Hint; solve for x). Answer: 22.5 = 0.9x; x = 22.5/0.9 = 25 songs
Homework. Homework problems should be more of the examples used in the lesson. Students should determine whether a relation is a function or not and give th reason why. They should get practice with the f(x) notation as in the examples. Finally, students should also write functions to represent particular situations as in Problem 4 of the examples above.