Integrated Math 2–3 Honors — Functions, Domain and Range, and Absolute-Value Equations
Corrected lesson transcript


## 0:00 — What Is a Function?

[0:04] A relation is a set of ordered pairs.
[0:21] Ok, we got a new definition. A function is a relation.
[0:45] So it's a relation with a condition that must be fulfilled.
[0:50] The condition is each x coordinate.
[1:02] Sometimes we call that the input
[1:05] So, function, relation, set each x coordinate corresponds to only one y coordinate.
[1:12] Okay.

## 1:13 — Functions from Ordered Pairs

[1:13] Are A and B both relations?
[1:18] Oh, it's not supposed to be a hard question.
[1:21] Yeah.
[1:21] They are what?
[1:23] Because they are sets of ordered pairs.
[1:24] They're supposed to just set the work.
[1:31] Is A a function?
[1:33] Is B a function?
[1:34] Are they both functions or neither of them functions?
[1:37] don't you discuss
[1:38] Is C a function?
[1:41] No.
[1:41] No, why not?
[1:43] Because 2 has 2 y-coordinates.
[1:46] Because the x-coordinate of 2 corresponds to 5 and corresponds to 8.
[1:58] Here in A, 1 corresponds to 3, 5 corresponds to 3.
[2:03] We don't care as long as the same x value only corresponds to one y value.
[2:09] So when we're looking at the list, if it's in roster notation like this, all you have
[2:15] to do is check.
[2:16] As long as it doesn't repeat an x value, it's a function, guaranteed.
[2:20] A repeated x-value means the relation is not a function.

## 2:24 — Graphs and the Vertical-Line Test

[2:25] This is the same as relation A, just the graph of the set.
[2:38] Relation C contains (2, 5), (2, 8), and (6, 3).
[2:48] This is relation C. It contains three ordered pairs.
[2:52] And I think you guys learned last year a test to see
[2:55] whether a relation is a function.
[2:57] What is it?
[2:58] line test.
[2:59] Called the vertical line test.
[3:03] How many remember that from last year?
[3:07] If every vertical line only touches it once, it is a function.
[3:14] Or if any vertical line touches the relation twice, it's not a function.
[3:21] Two ways to say the vertical line test.
[3:25] Let's write that down.
[3:38] If any vertical line touches the relation twice, it
[3:50] Is this a relation?
[3:52] Yes.
[3:53] Yeah, the set of ordered pairs is some linear relation.
[4:00] Is this also a function?
[4:02] Yes.
[4:02] Yes, and how do we know?
[4:04] Because every x-coordinate has only one y-coordinate.
[4:07] Either by the definition, every x-coordinate has only one y-coordinate, or corresponds only one y-coordinate, or?
[4:13] Passes the vertical line.
[4:14] Passes the vertical line test. Every single vertical line touches it most once.
[4:19] Okay.
[4:26] We're going to learn in a couple months about circles.
[4:32] Is this a relation?
[4:34] Yes.
[4:36] Is this a set of ordered pairs?
[4:38] Yeah.
[4:40] Anything on a coordinate plane in black here is just a set of ordered pairs.
[4:47] It is a relation.
[4:48] Is it a function?
[4:49] No.
[4:50] No.
[4:55] At least one vertical line touches it twice, so it is not a function.
[4:59] We can still do lots of math with relations.
[5:02] We'll learn about that later.
[5:04] But just distinguishing what is a function, what is not a function.

## 5:08 — Domain and Range

[5:08] OK, another definition.
[5:09] And the domain of a relation or function is the set
[5:27] is the range
[5:35] The range of a relation is the set of all y-coordinates.
[5:48] Sometimes we say the domain is the set of all inputs and the range is the set of all outputs.
[5:53] So what's the domain of A?
[5:58] 1, 4, 5.
[5:59] 1, 4, 5.
[6:01] Close.
[6:04] It's the set containing 1, 4, 5.
[6:08] These are sets of numbers.
[6:14] And the range?
[6:20] of three and six.
[6:26] We've got some linear function.
[6:33] What is the domain of this?
[6:39] And how can we figure that out?
[6:45] It is all real numbers.
[6:47] Why is that?
[6:50] This is going down and to the left.
[6:53] So anywhere on the x-coordinate or x-axis.
[6:56] If you go down you will eventually have a y-coordinate.
[7:00] And let's say this is y equals x plus 5.
[7:06] I'm just making up.
[7:07] It could be something like that.
[7:09] Okay, you can plug in any real number for x and always get a y coordinate.
[7:15] So the domain is all real numbers.
[7:17] could either write it like this.
[7:21] Or in interval notation it would be?
[7:26] Negative infinity to infinity.
[7:33] And what about the range?
[7:45] This goes forever to the right, but it's also going forever up and it's going forever to the left and down.
[7:51] So if it takes on all y values down and all y values up, so the range is also
[8:01] What's the domain of this?
[8:06] Look left and right, it goes forever to the right, forever to the left.
[8:10] Every x coordinate has a y coordinate.
[8:14] All real numbers. And what is the range? It only has one number in the range. The set containing four.

## 8:31 — Domain and Range of Common Graphs

[8:33] We're going to learn something later called quadratic functions.
[8:36] This is the example of a quadratic function.
[8:42] What is the domain of this? All real numbers. It goes forever. Even it's going
[8:54] out very fast, it goes forever to the right. There's always a y coordinate.
[8:58] The equation for this is y equals x squared. You can plug in any x value, you always
[9:03] get a y value. However, what is the range? All real numbers greater than or equal to
[9:10] zero. All real numbers greater than or equal to zero. That's true. It never has any negative
[9:24] values but it's definitely the set of and how do we write that down zero to
[9:38] infinity do I include zero yeah it's on the origin and do we include infinity
[9:48] Never.
[9:50] Ok.
[9:53] This has a smallest y value.
[9:55] smallest y value is this coordinate right here.
[10:02] Ok and the range has a minimum.
[10:05] The minimum is zero.
[10:07] Ok.
[10:08] If the function was like this, there would be a maximum.
[10:13] and we take on all of these y values to wherever that point is.
[10:19] so relations, domain range, vertical line test, some examples.
[10:28] Alright, new topic.

## 10:31 — Absolute Value and Its Piecewise Definition

[10:32] What is the absolute value of the number 5?
[10:36] 5.
[10:37] 5.
[10:38] What is the absolute value of the number negative 5?
[10:42] 5.
[10:43] 5.
[10:46] What is the absolute value of x?
[10:50] x is it x?
[10:59] What if x is negative 5?
[11:05] that true?
[11:06] No.
[11:07] So it's not x.
[11:10] Uh oh.
[11:10] What is the absolute value of x?
[11:17] What's that?
[11:19] So is it both?
[11:27] Let's try that with this number. Is the absolute value 5 plus or minus 5?
[11:33] No, it's just the plus. I like the idea though.
[11:41] Any other ideas?
[11:54] Does it depend?
[11:57] it depends on the value of x.
[12:01] If x is positive, it's x.
[12:05] What if x is negative?
[12:08] is negative x.
[12:12] The absolute value of x is what we call a piecewise-defined function.
[12:17] defined function. How did they define in elementary school? Oh, how many spaces is from the origin?
[12:32] Yes, it is the distance from the origin.
[12:39] To describe that distance with a formula,
[12:51] We need something better
[12:54] Okay, so because it depends we have to break it up into his two cases
[12:59] So the way we do that is with a it's called a piecewise defined function
[13:05] Did you guys learn this last year
[13:12] So the absolute value of x is x if what's true about x?
[13:19] If x is positive.
[13:21] If x is positive or 0.
[13:26] So you can write it like this.
[13:27] if x is greater than or equal to 0.
[13:34] And then if x is less than 0, what is the absolute value?
[13:42] It's negative x. It's the opposite of the value of x.
[13:52] This is the, called the piecewise definition of the absolute value of x.
[14:15] and sometimes we write it like this
[14:22] its x
[14:24] if x is contained in
[14:28] 0 to infinity
[14:33] first negative x if x
[14:37] is in the set negative infinity to zero.
[14:49] This symbol, ∈, is read as ‘is an element of.’
[14:55] It's the Greek letter, lowercase Greek letter epsilon.
[15:02] Okay.
[15:05] It simply means in math when we're writing down symbols,
[15:08] It means ‘is an element of,’
[15:16] so if x is contained in the set.
[15:20] We use a lot in set notation.
[15:27] Okay, and again, when we see one girly bracket,
[15:29] this doesn't mean a set.
[15:31] This is how we write a piecewise defined function.
[15:36] You list all the different pieces, all the different intervals.
[15:41] You can either write interval notation or you can write inequalities like I did up here.
[15:49] And so on.

## 15:50 — Solve |x| = 7

[15:51] Now, what's this thing I bring up here?
[15:55] I actually buy it.
[15:56] That's the neatest thing.
[15:58] Is it an expression or an equation?
[16:01] I call this an absolute value equation because it contains an absolute value.
[16:08] Is it true, false or open?
[16:10] Open.
[16:10] Open.
[16:11] Are the solutions numbers or are they ordered pairs?
[16:14] Numbers.
[16:15] Numbers because?
[16:16] One variable.
[16:17] There is only one variable.
[16:21] We're going to solve this.
[16:23] What's it mean to solve?
[16:26] How
[16:27] so
[16:29] if x
[16:30] is less than or equal to, sorry, less than 0,
[16:42] What does the absolute value of x equal?
[16:49] Hold on.
[16:51] Since x is less than 0, are we here or are we here?
[16:59] If x is less than 0, we're here.
[17:01] Yeah.
[17:04] OK.
[17:07] So the absolute value of x is equal to this guy.
[17:15] So I replace this with negative x.
[17:19] And now I no longer have an absolute value equation.
[17:23] It's got a simple linear equation.
[17:26] And we multiply it by negative 1.
[17:29] and we get x equals negative 7.
[17:32] One of our solutions is going to be negative 7.
[17:37] And then case two, if x is greater than or equal to 0,
[17:46] what does the absolute value of x equal to?
[17:49] 7.
[17:51] No.
[17:51] The absolute value equals x in this case.
[18:02] this okay and then of course the solution here is seven so the solutions
[18:12] The solutions to the absolute-value equation are both values.
[18:30] sometimes we write this as plus or minus seven these mean the same thing it
[18:41] really means both positive 7 and negative 7.
[18:47] OK, we solved the first absolute

## 18:49 — Isolate the Absolute Value First

[18:49] Okay, before we get to the definition, what are we going to do first?
[18:58] Add 4.
[19:01] So, yeah, get the absolute value all by itself.
[19:09] Got to get the absolute value all by itself.
[19:12] Add 4, we got 10.
[19:16] Divide by 2.
[19:19] Okay.
[19:21] And then we split it up into two cases.
[19:36] Oops.
[19:47] So, brain-run equals zero, absolute value of x is x, solution's five.
[20:08] Once we get to here, is it always going to be plus or minus that number?
[20:20] It is.
[20:21] Okay.
[20:28] So if any time you have a
[20:36] If absolute value of x equals some number a
[20:41] and a is greater than or equal to zero
[20:47] Then x will equal plus or minus a.
[20:56] x will equal
[20:59] yeah let's just do that plus or minus a.
[21:08] Okay and then we get
[21:09] which is two equations x equals a and x equals negative a.
[21:14] we get plus or minus A.

## 21:18 — Expressions Inside Absolute Value

[21:18] Now we have something inside the absolute value.
[21:26] Sorry—inside the absolute value.
[21:29] What are we going to do with this?
[21:32] We'll just make this a...
[21:41] I think of this as my x now.
[21:47] Once we get the absolute value all by itself, just write x minus 5 equals 7 or x minus 5
[21:59] equals negative 7.
[22:02] Same thing as x minus 5 equals plus or minus 7, which is two separate equations.
[22:09] Once we get the two separate equations we're going to solve them separately.
[22:28] We got two solutions.
[22:30] We typically write them in numerical order but it really doesn't matter.
[22:34] the order of a set does not matter

## 22:37 — A Multi-Step Absolute-Value Equation

[22:37] Let me change this.
[23:06] First step.
[23:25] And
[23:26] And once we get the absolute value by itself, we have to write two separate equations.
[23:33] 5x minus 10 equals 1, or 5x minus 10 is negative 1.
[23:51] Now we just got two simple linear equations.
[23:55] Add ten plus ten.
[24:29] I have two solutions.
[24:37] Okay, so for all these,
[24:39] once you get the absolute value by itself,
[24:43] just write two separate equations.
[24:46] It'll either equal positive that number
[24:48] or negative that number.
[24:51] Sound good?
[24:53] Yeah.
[24:54] Okay.
[24:59] I think you'll be able to do the homework.
[25:01] So we're gonna stop here.