WEBVTT mathematics/algebra-2/eaton 00:00:00.000 --> 00:00:02.300 Welcome to Educator.com. 00:00:02.300 --> 00:00:07.800 For today's Algebra II lesson, we are going to be discussing relations and functions. 00:00:07.800 --> 00:00:13.100 And recall that some of these concepts were discussed in Algebra I, so this is a review. 00:00:13.100 --> 00:00:20.800 And if you need a more detailed review, check out the Algebra I lectures here at Educator. 00:00:20.800 --> 00:00:31.200 Beginning with the concept of the coordinate plane: the coordinate plane describes each point as an ordered pair of numbers (x,y). 00:00:31.200 --> 00:00:36.200 The first number is the x-coordinate, and the second is the y-coordinate. 00:00:36.200 --> 00:00:57.500 For example, consider the ordered pair (-4,-2): this is describing a point on the coordinate plane with an x-coordinate of -4 and a y-coordinate of -2. 00:00:57.500 --> 00:01:13.500 Or the pair (0,2): the x-value would be 0, and the y-value would be 2; this is the point (0,2) on the coordinate plane. 00:01:13.500 --> 00:01:20.300 Or (3,5): x is 3; y is 5. 00:01:20.300 --> 00:01:37.000 Also, recall that the quadrants are labeled with the Roman numerals: I, and then (going counterclockwise) quadrant II, quadrant III, and quadrant IV. 00:01:37.000 --> 00:01:42.900 In the coordinate pairs in the first quadrant, the x is positive, as is the y. 00:01:42.900 --> 00:01:54.200 In the second quadrant, you will have a negative value for x and a positive value for y, such as (-2,4)--that would be an example. 00:01:54.200 --> 00:02:07.300 In the third quadrant, both x and y are negative; and then, in the fourth quadrant, x is positive; y is negative. 00:02:07.300 --> 00:02:14.800 And we will be using the coordinate plane frequently in these lessons, in order to graph various equations. 00:02:14.800 --> 00:02:19.800 Recall that a relation is a set of ordered pairs. 00:02:19.800 --> 00:02:28.800 The domain of the relation is the set of all the first coordinates, and the range is the set of all the second coordinates. 00:02:28.800 --> 00:02:38.500 A relation is often written as a set of ordered pairs, using braces to denote that this is a set, 00:02:38.500 --> 00:02:52.000 and then the ordered pairs, each in parentheses, separated by a comma. 00:02:52.000 --> 00:03:11.900 Sometimes, the relation is represented as a table; so, -2, 1; -1, 0; 0, 1; and 1, 2. 00:03:11.900 --> 00:03:16.300 We will be doing some graphing of relations also, in just a little bit. 00:03:16.300 --> 00:03:27.400 So, as discussed up here, the domain is the set of all first coordinates. 00:03:27.400 --> 00:03:38.800 And the set of all first coordinates here would be {-2, -1, 0, 1}. 00:03:38.800 --> 00:03:52.500 The range is the set of the second coordinates; so the range right here--all the y-values--is {1, 0, 1, 2}. 00:03:52.500 --> 00:04:05.500 However, you don't actually need to write the 1 twice; so in actuality, it would be written as such. 00:04:05.500 --> 00:04:21.300 It is OK to repeat the values if you want, but usually, we just write each value in the domain or range once; each is represented once. 00:04:21.300 --> 00:04:32.000 Functions are a certain type of relation: so, all functions are relations, but not all relations are functions. 00:04:32.000 --> 00:04:40.200 A function is a relation in which each element of the domain is paired with exactly one element of the range. 00:04:40.200 --> 00:04:58.300 For example, consider the relation shown: {(1,4), (2,5), (3,8), (4,10)}. 00:04:58.300 --> 00:05:04.900 Each member of the domain corresponds to exactly one element of the range. 00:05:04.900 --> 00:05:14.400 We don't have a situation where it is saying {(1,4), (1,6), (1,8)}, where that member of the domain is paired with multiple members of the range. 00:05:14.400 --> 00:05:20.900 Another way, again, to represent this as a table--another method that can be used--is mapping. 00:05:20.900 --> 00:05:26.700 And mapping is a visual device that can help you to determine if you have a function or not, 00:05:26.700 --> 00:05:33.500 by showing how each element of the domain is paired with an element of the range. 00:05:33.500 --> 00:05:47.400 A map would look something like this: over here, I am going to put the elements of the domain, 1, 2, 3, and 8; 00:05:47.400 --> 00:05:55.400 over here, the elements of the range: 4, 5, 8, and 10. 00:05:55.400 --> 00:05:59.700 And then, using arrows, I am going to show the relationship between the two. 00:05:59.700 --> 00:06:14.600 So, 1 corresponds to 4 (or is paired with 4); 2 to 5; 3 to 8; and (this should actually be 4) 4 to 10. 00:06:14.600 --> 00:06:23.000 OK, so as you can see, there is only one arrow going from each element of the domain to each element of the range. 00:06:23.000 --> 00:06:26.300 And that tells me that I do have a function. 00:06:26.300 --> 00:06:32.400 Let's look at a different situation, using a table form: let's look at a second relation. 00:06:32.400 --> 00:06:43.500 In this one, I am going to have (-2,2), (-3,2), (-4,5), and (-6,7). 00:06:43.500 --> 00:06:56.300 And I am going to go ahead and map this: -2, -3, -4, and -6: these are my elements of the domain. 00:06:56.300 --> 00:07:02.100 For the range, I don't have to write 2 twice; I am just going to write it once; 5, and 7. 00:07:02.100 --> 00:07:15.800 OK, -2 corresponds to 2; -3 also corresponds to 2; -4 corresponds to 5; and -6 corresponds to 7. 00:07:15.800 --> 00:07:24.900 This is also a function, so both of these are relations, and they are also functions. 00:07:24.900 --> 00:07:32.500 It is OK for two elements of the domain to be paired with the same element of the range; this is allowed. 00:07:32.500 --> 00:07:47.700 What is not allowed is if I were to have a situation where I had {1, 2, 3}, {4, 5, 6}; and I had 1 paired with 4, and 1 paired with 5. 00:07:47.700 --> 00:07:56.200 So, if you have two arrows coming off an element of the domain, then this is not a function. 00:07:56.200 --> 00:08:01.800 Here are two examples of relations that are also functions. 00:08:01.800 --> 00:08:06.100 There is a specific type of function that is called a one-to-one function. 00:08:06.100 --> 00:08:13.700 And a function is one-to-one if distinct elements of the domain are paired with distinct elements of the range. 00:08:13.700 --> 00:08:21.400 In the previous example, we saw a situation where we did have a one-to-one function, and another situation where we did not. 00:08:21.400 --> 00:08:39.400 OK, so to review: the ordered pairs in that first function that we just discussed were (1,4), (2,5), (3,8), and (4,10). 00:08:39.400 --> 00:08:49.600 OK, and we can use mapping, again, to determine what the situation is with this relation (which is also a function). 00:08:49.600 --> 00:08:59.800 The domain is {1, 2, 3, 4}; and the range is {4, 5, 8, 10}. 00:08:59.800 --> 00:09:12.300 When I put my arrows to show this relationship, you see that distinct elements of the domain are paired with distinct elements in the range. 00:09:12.300 --> 00:09:18.400 1 is paired with 4; they are each unique--each pair is unique. 00:09:18.400 --> 00:09:47.600 Looking at the other function that we discussed: the pairs are (-2,3), (-3,2), (-4,3)...slightly different, but the same general concept...slightly different, though. 00:09:47.600 --> 00:10:04.700 OK, here I have -2, -3, -4, and -6; over here, in the range, I have 3, 2...I am not going to repeat the 3--I already have that...and then 7. 00:10:04.700 --> 00:10:20.300 -2 corresponds to 3; -3 corresponds to 2; -4 also corresponds to 3; -6 corresponds to 7. 00:10:20.300 --> 00:10:28.500 This is still a function; OK, so these are both functions: function, function. 00:10:28.500 --> 00:10:39.500 However, this is a one-to-one function; this is not one-to-one. 00:10:39.500 --> 00:10:48.000 They are both functions, since each element of the domain is paired only with one element of the range. 00:10:48.000 --> 00:10:56.300 But in this case, it is not a unique element of the range: these two, -2 and -4, actually share an element of the range. 00:10:56.300 --> 00:11:01.900 In other words, this is unique; it is a one-to-one correspondence. 00:11:01.900 --> 00:11:13.300 OK, we can graph relations and functions by plotting the ordered pairs as points in the coordinate plane, as discussed a little while ago. 00:11:13.300 --> 00:11:15.700 There are a couple of types of graphs that you can end up with. 00:11:15.700 --> 00:11:22.300 The first is discreet, and the second is continuous; let's look at those two different types. 00:11:22.300 --> 00:11:38.200 Consider this relation: OK, so if I am asked to graph this relation, I am going to graph each point: 00:11:38.200 --> 00:11:53.900 (-4,-2): that is going to be right here; (-2,1)--right here; here, (0,2), 0 on the x, 2 on the y. 00:11:53.900 --> 00:12:01.400 This is a discrete function--discrete graph--discrete relation. 00:12:01.400 --> 00:12:07.600 This actually is both a relation and a function; so it is a discrete relation or a discrete function. 00:12:07.600 --> 00:12:13.700 And the reason is because I have a set of discrete points; they are not connected. 00:12:13.700 --> 00:12:20.800 And I can't connect them, because I haven't been given anything in between, or a way to know if or what lies in between these. 00:12:20.800 --> 00:12:26.800 I can't just connect them when I don't know; there could be a point up here, or actually this is just the entire relation. 00:12:26.800 --> 00:12:29.100 So, I can just work with what is given. 00:12:29.100 --> 00:12:39.100 OK, a different scenario would be if I am given a relation y=x+1. 00:12:39.100 --> 00:12:58.900 And I can go ahead and plot this out, if I say, "OK, when x is -1, -1+1 is 0; when x is 0, y is 1; when x is 1, 1+1 is 2; when x is 2, y is 3." 00:12:58.900 --> 00:13:01.300 OK, so I am going to go ahead and plot this out. 00:13:01.300 --> 00:13:10.400 When x is -1, y is 0; when x is 0, y is 1; when x is 1, y is 2. 00:13:10.400 --> 00:13:13.200 Let's remove this out of the way. 00:13:13.200 --> 00:13:18.000 When x is 1, y is 2; when x is 2, y is 3. 00:13:18.000 --> 00:13:21.200 Now, I have a set of points, because these are the points I chose. 00:13:21.200 --> 00:13:26.800 But because I am given this equation, there is an infinite number of points in between. 00:13:26.800 --> 00:13:40.700 I could have chosen an x of .5 to get the value 1.5 here, to fill that in--and on and on, until this becomes continuous and forms a line. 00:13:40.700 --> 00:13:53.700 So, this is a continuous function: the graph is a connected set of points, 00:13:53.700 --> 00:14:01.100 so the relation (or the function in this case, since we do have a function) is continuous relation or continuous function, 00:14:01.100 --> 00:14:09.000 because I have a line; whereas this, which is a set of points, is a discrete relation. 00:14:09.000 --> 00:14:16.600 OK, one visual way to tell if a relation is a function is using the vertical line test. 00:14:16.600 --> 00:14:26.500 And a relation is a function if and only if no vertical line intersects its graph at more than one point. 00:14:26.500 --> 00:14:30.100 This is most easily understood through just working through an example. 00:14:30.100 --> 00:14:33.900 Consider if you were given the following graph. 00:14:33.900 --> 00:14:40.400 OK, the vertical line test: what you are seeing is, "Can you put a vertical line 00:14:40.400 --> 00:14:47.600 somewhere on the graph so that it intersects the graph at more than one point?" 00:14:47.600 --> 00:14:56.900 And I can: I put a vertical line here, and it intersects this graph at 1, 2, 3 places. 00:14:56.900 --> 00:15:03.200 Over here, it only intersects at one place; that is fine; but if I can draw a vertical line anywhere on the graph 00:15:03.200 --> 00:15:11.800 that intersects at more than one place, then we say that this failed the vertical line test. 00:15:11.800 --> 00:15:19.000 And when something fails the vertical line test, it means that it is not a function. 00:15:19.000 --> 00:15:26.500 The reason this works is that, if two or three or more points share the same x-value, 00:15:26.500 --> 00:15:31.500 then they are going to lie directly above or below each other on the coordinate plane. 00:15:31.500 --> 00:15:40.200 For example, looking right here, I have x = 3; x is 3; y is 0. 00:15:40.200 --> 00:15:50.200 Then, I look right above it, up here: again, x is 3, and y is...say 2.1--pretty close. 00:15:50.200 --> 00:15:58.800 Then, I look up here; again, x is 3, and y is about 4.6, approximately. 00:15:58.800 --> 00:16:07.900 So, when x-values are the same, but then the y-values are different, that is telling me 00:16:07.900 --> 00:16:15.100 that members of the domain are paired with more than one member of the range; by definition, that is not a function. 00:16:15.100 --> 00:16:24.400 OK, consider a different graph--consider a graph like this of a line--a straight line. 00:16:24.400 --> 00:16:36.600 OK, now, anywhere that I pass a vertical line through--anywhere on this graph--it is only going to intersect at one point. 00:16:36.600 --> 00:16:46.200 So, this passed the vertical line test. 00:16:46.200 --> 00:16:53.700 Therefore, this line, this graph, represents a function. 00:16:53.700 --> 00:17:04.100 OK, so the vertical line test is a visual way of determining if a relation is a function. 00:17:04.100 --> 00:17:11.300 Working with equations: an equation can represent either a relation or a function. 00:17:11.300 --> 00:17:16.300 If an equation represents a function, then there is some terminology we use. 00:17:16.300 --> 00:17:23.100 And let's start out by just looking at an equation that represents a function. 00:17:23.100 --> 00:17:29.900 The variable corresponding to the domain is called the independent variable, and the other variable is the dependent variable. 00:17:29.900 --> 00:17:37.500 So, here I have x and y; and let's look at some values--let's let x be -1. 00:17:37.500 --> 00:17:43.600 Well, -1 times 2 is -2, minus 1--that is going to give me -3. 00:17:43.600 --> 00:17:50.700 When x is 0, 0 times 2 is 0, minus 1 is -1. 00:17:50.700 --> 00:17:57.200 When x is 1, 1 times 2 is 2, minus 1--y is 1. 00:17:57.200 --> 00:18:04.600 When x is 2, 2 times 2 is 4, minus 1 gives me 3. 00:18:04.600 --> 00:18:19.400 So, looking at how this worked, x is the independent variable. 00:18:19.400 --> 00:18:29.000 The value of x is independent of y; I am just picking x's, and here it could be any real number. 00:18:29.000 --> 00:18:36.400 We sometimes also say that this is the input; and the reason is that I pick a value for x (say 0), 00:18:36.400 --> 00:18:47.400 and I put it in--I input it into the equation; then, I do my calculation, and out comes a y-value. 00:18:47.400 --> 00:19:01.600 So, the value of y is dependent on x; therefore, it is the dependent variable; and we also sometimes say that it is the output. 00:19:01.600 --> 00:19:11.100 You put x in and do the calculation; out comes the value of y; so x is independent, and y is dependent. 00:19:11.100 --> 00:19:16.900 The notation that you will see frequently in algebra is function notation. 00:19:16.900 --> 00:19:24.900 We have been writing functions like this: y = 4x + 3; but you will often see... 00:19:24.900 --> 00:19:30.600 instead of an equation written like this, if it is a function, you will see it written as such. 00:19:30.600 --> 00:19:39.100 And when we say this out loud, we pronounce it "f of x equals 4x plus 3." 00:19:39.100 --> 00:19:46.600 And we are talking about the value of a function for a particular value of x. 00:19:46.600 --> 00:19:53.400 So, we say, "The function of f at a particular x." 00:19:53.400 --> 00:20:04.100 Let's let x equal 3; then, we can talk about f of 3--the value of the function, the value of y, 00:20:04.100 --> 00:20:09.200 of the dependent variable, when the independent variable, x, is 3. 00:20:09.200 --> 00:20:17.400 And in that case, since it is telling us that x is 3, I am going to substitute in 3 wherever there is an x. 00:20:17.400 --> 00:20:31.100 And I could calculate that out to tell me that f(3) is...4 times 3 is 12, plus 3...so f(3) is 15. 00:20:31.100 --> 00:20:46.400 Here, x is an element of the domain, of the independent variable; f(x) is an element of the range. 00:20:46.400 --> 00:20:53.000 So again, we are going to be using this function notation throughout the remainder of the course. 00:20:53.000 --> 00:21:00.600 Looking at the first example: the relation R is given by this set of coordinate pairs. 00:21:00.600 --> 00:21:07.500 Give the domain and range, and determine of R is a function. 00:21:07.500 --> 00:21:15.600 Well, recall that the domain is comprised of the first element of each of these coordinate pairs. 00:21:15.600 --> 00:21:31.100 So in this case, the domain would be {1, 2, 6, 5, 7}. 00:21:31.100 --> 00:21:43.100 The range: the range is comprised of the second element of each ordered pairs, so I have 4, 3...4 again; 00:21:43.100 --> 00:21:49.500 I don't need to write that again; 3--I have 3 already; and 5; I am just writing down the unique elements. 00:21:49.500 --> 00:21:51.400 This is the domain, and this is the range. 00:21:51.400 --> 00:22:00.000 Now, is R a function? Well, I can always use mapping to just help me determine that. 00:22:00.000 --> 00:22:10.500 And I am going to write down my members of the domain, and my elements of the range. 00:22:10.500 --> 00:22:13.500 And then, I am going to use arrows to show the correspondence between each: 00:22:13.500 --> 00:22:29.800 1 and 4--1 is paired with 4; 2 is paired with 3; 6 is also paired with 4; 5 is paired with 3; and 7 is paired with 5. 00:22:29.800 --> 00:22:34.200 Now, I am looking, and I only see one arrow leading from each element of the domain. 00:22:34.200 --> 00:22:39.200 There is no element of the domain that is paired with two elements of the range. 00:22:39.200 --> 00:22:52.700 So, in this case, this is a function; so, is R a function? Yes, this relation is a function--R is a function. 00:22:52.700 --> 00:22:55.400 So, always double-check and make sure you have answered each part. 00:22:55.400 --> 00:23:04.500 I found the domain; I found the range; and I determined that R is a function. 00:23:04.500 --> 00:23:12.900 OK, the relation R is given by the equation y=2x²+4; is R a function? 00:23:12.900 --> 00:23:20.100 What are the domain and range? Is R discrete or continuous? 00:23:20.100 --> 00:23:32.500 Let's just look at some values for x and y to help us determine if this relation is a function. 00:23:32.500 --> 00:23:42.100 If I let x equal -1, -1 times -1 is 1, times 2 is 2, plus 4 is 6. 00:23:42.100 --> 00:23:48.400 OK, if x is 0, this is 0, plus 4--that gives me 4. 00:23:48.400 --> 00:23:54.600 If x is 3, 3 squared is 9, times 2 is 18, plus 4 is 22. 00:23:54.600 --> 00:24:10.000 So, as you are going along, you can see that, for any value of x, there is only one value of y; therefore, R is a function. 00:24:10.000 --> 00:24:26.700 What is the domain? Well, I could pick any real number for an x-value that I wanted, so the domain is all real numbers. 00:24:26.700 --> 00:24:35.200 You might, at first glance, say, "Oh, the range is all real numbers, as well"; but that is not correct, because look at what happens. 00:24:35.200 --> 00:24:46.300 Because this is x², whenever I have a negative number, it becomes positive; if I have a positive number, it stays positive, of course. 00:24:46.300 --> 00:24:52.800 Therefore, if I have, say, -1, that becomes 1; this becomes 6. 00:24:52.800 --> 00:25:04.800 So, I am not going to get any value lower than...for y, the smallest value I will get is for when x is 0. 00:25:04.800 --> 00:25:10.600 OK, so if x is 0, y is 4; because -1 is going to give me a bigger value--it is going to give me 6. 00:25:10.600 --> 00:25:15.800 If I do -2, that is going to be 4 times 2 is 8, plus 4 is 12. 00:25:15.800 --> 00:25:21.800 So, the lowest value that I will be able to get for y will occur when x is 0. 00:25:21.800 --> 00:25:25.200 And that is going to give me a y-value of 4. 00:25:25.200 --> 00:25:30.600 Therefore, the range is that y is greater than or equal to 4. 00:25:30.600 --> 00:25:38.600 So, the most difficult part of this was just realizing that the range is not as broad as it looked initially. 00:25:38.600 --> 00:25:45.600 Because this involves squaring a number, there is a limit on how low you are going to go with the y-value. 00:25:45.600 --> 00:25:58.900 So, this is a range with a domain of all real numbers, and a range of greater than or equal to 4. 00:25:58.900 --> 00:26:12.400 OK, in Example 3, graph the relation R given by 2x - 4y = 8. 00:26:12.400 --> 00:26:22.900 Is R a function? Find its domain and range. Is R discrete or continuous? 00:26:22.900 --> 00:26:30.600 OK, so graph the relation given by 2x - 4y = 8. 00:26:30.600 --> 00:26:38.100 Let's go ahead and find some x and y values, so that we can graph this. 00:26:38.100 --> 00:26:47.100 When x is 0, we need to be able to solve for y; when x is 0, let's figure out what y is. 00:26:47.100 --> 00:26:55.500 0 - 4y equals 8; therefore, y equals -2 (dividing both sides by -4). 00:26:55.500 --> 00:27:14.700 OK, when x is 2, 2 times 2 minus 4y equals 8; that is 4 minus 4y equals 8; that is -4y equals 4; y = -1. 00:27:14.700 --> 00:27:26.200 And let's do one more: when x is -2, this is going to give me -4 - 4y = 8. 00:27:26.200 --> 00:27:37.000 That is going to then give me, adding 4 to both sides, -4y = 12, or y = -3; that is good. 00:27:37.000 --> 00:28:06.200 All right, so when x is 0, y is -2; when x is 2, y is -1; when x is -2, y is -3, right here. 00:28:06.200 --> 00:28:11.100 I am asked to graph it; and I have some points here that I generated, 00:28:11.100 --> 00:28:22.000 but I also realize that I could have picked points in between these, which would actually end up connecting this as a line. 00:28:22.000 --> 00:28:27.500 So, I am not just given a set of ordered pairs; I am given an equation that could have an infinite number of values for x, 00:28:27.500 --> 00:28:33.400 which would allow me to graph this as a continuous line. 00:28:33.400 --> 00:28:40.400 Therefore, I graphed the relation...is R a function? 00:28:40.400 --> 00:28:51.300 Is R discrete or continuous? Well, I have already answered that--seeing the graph of this, I know that this is continuous. 00:28:51.300 --> 00:29:06.100 And let's see, the next step: is R a function? 00:29:06.100 --> 00:29:17.700 Yes, it is a function, because if I look, for every value of x (for every value of the domain), there is one value only of the range. 00:29:17.700 --> 00:29:21.900 So, every element of the domain is paired with only one element of the range. 00:29:21.900 --> 00:29:25.600 It is continuous, and it is a function. 00:29:25.600 --> 00:29:37.900 Find the domain and range: well, this is another case where I could choose x to be any real number, so it would be all real numbers--any real number. 00:29:37.900 --> 00:29:44.400 Here, the situation is the same for the range--all real numbers. 00:29:44.400 --> 00:29:52.300 Depending on my x-value, I could come up with infinite possibilities for what the range would be, what the y-value would be. 00:29:52.300 --> 00:30:05.500 R is a function; its domain and range are all real numbers; and this is a continuous function. 00:30:05.500 --> 00:30:24.500 OK, in Example 4, we are given f(x) = 3x² - 4, and asked to find f(2), f(6), and f(2k). 00:30:24.500 --> 00:30:33.800 First, f(2): recall that, when you are asked to find a function for a particular value of x, 00:30:33.800 --> 00:30:51.900 you simply substitute that value for x in the equation; so f(2) equals 3(4) - 4, so that is 12 - 4; so f(2) = 8. 00:30:51.900 --> 00:31:00.100 Next, I am asked to find f(6), and that is going to equal 3(6²) - 4. 00:31:00.100 --> 00:31:17.600 f(6) = 3(36) -4, and that turns out to be 108 - 4, so f(6) is 104. 00:31:17.600 --> 00:31:27.300 Now, at first, this f(2k) might look kind of difficult; but you treat it just the same as you did with the numbers, when x is a numerical value. 00:31:27.300 --> 00:31:31.200 Everywhere I see an x, I am going to insert 2k. 00:31:31.200 --> 00:31:42.000 And figuring this out, 2 times 2, 2 squared, is 4; k times k is k². 00:31:42.000 --> 00:31:50.300 3 times 4 is 12, so I have 12k² - 4; so f(2k) = 12k² - 4. 00:31:50.300 --> 00:31:59.400 So again, if you are asked to find the function of a particular value of x, you simply substitute whatever is given, including variables, for x. 00:31:59.400 --> 00:32:05.000 That concludes this lesson of Educator.com; I will see you back here soon!