WEBVTT 00:21.520 --> 00:26.660 In this video, we're going to learn about functions. We'll begin our discussion 00:26.720 --> 00:35.002 with relations, domain, and range. In a previous section, we learned that we can write an equation 00:35.040 --> 00:41.876 to discuss the relationship between two variables. For example in a square, 00:41.920 --> 00:49.383 if we let the length of one of the sides of a square equal x and the area of that square 00:49.440 --> 00:56.590 equal y, then I can write an equation that demonstrates the relationship of those variables. 00:56.640 --> 01:04.198 The area of the square y is equal to the length of the sides squared, 01:04.240 --> 01:14.675 or x squared. So this equation y equals x squared shows a relationship between these two variables. 01:14.720 --> 01:23.617 A relation is a set of ordered pairs. A set of ordered pairs is called a relation. 01:23.680 --> 01:30.257 The set of all of the x-coordinates is the domain of that relation, and the set of the 01:30.320 --> 01:39.166 y-coordinates is called the range of a relation. Now equations such as this y equals x squared 01:39.200 --> 01:49.376 are also called relations since equations in two variables define a set of ordered pairs. 01:49.440 --> 02:01.222 A function is a special type of relation. A function is a set of ordered pairs that assigns to each x-value exactly 02:01.280 --> 02:11.699 one y-value. In just a moment, we're going to explore this definition a little further. 02:11.760 --> 02:18.906 In these two examples, we are going to state the domain and range of the given relation. 02:18.960 --> 02:26.380 And then we're going to determine if the relation is a function. Recall the definition of a function. 02:26.413 --> 02:38.025 A function is a set of ordered pairs that assigns to each x-value exactly one y-value. Let's take a look at these relations. 02:38.080 --> 02:46.500 Here's our set of ordered pairs. The domain is the set of the x-values of these ordered pairs. 02:46.560 --> 02:58.846 So in this case the domain is the set of numbers that contains four, negative three, 02:58.880 --> 03:06.520 and nine. Many times it's convenient to write these in ascending order, 03:06.560 --> 03:11.725 but the set of numbers is still the same regardless of the order that this is in. 03:11.760 --> 03:18.432 In any case, we do not duplicate a number when we write it in a set. So I will write this set 03:18.480 --> 03:35.482 as negative three, four, and nine. The range is the set of y-values in these ordered pairs. 03:35.520 --> 03:41.956 The y-values here are negative four. Here's another negative four. Remember 03:42.000 --> 03:53.233 we'll only write this once. And zero. So the range is the set negative four and zero. 03:53.280 --> 03:59.239 These are the domain and range. The next part of the instructions asks us to determine whether or 03:59.280 --> 04:07.147 not this relation is a function. So we need to determine if each element in the domain 04:07.200 --> 04:18.759 is paired with exactly one element in the range. Negative three is paired only with negative four. 04:18.800 --> 04:29.069 Four is paired only with negative four. And nine is paired only with zero. 04:29.120 --> 04:37.144 So even though the y-value negative four is repeated, each x-value is assigned to 04:37.200 --> 04:55.396 exactly one y-value. Therefore, this relation is a function. So we will say yes it is a function. 04:55.440 --> 05:03.871 Here's our next relation. Let's look at the domain. The domain is the set of x-values 05:03.920 --> 05:11.779 in our relation. We have the x-values zero, three, negative two, and zero 05:11.840 --> 05:23.056 is repeated. If I write these in ascending order, my domain is negative two, zero, and three. 05:23.120 --> 05:33.167 Remember, we only write the zero once, even though it shows up in two ordered pairs. 05:33.200 --> 05:43.644 The range is the set of the y-values in the relation. The y-values include the numbers five, 05:43.680 --> 05:52.920 zero, ten, and negative eight. Writing these in ascending order, we have negative eight, 05:52.960 --> 06:07.935 zero, five, and ten. Now let's determine whether or not this relation is also a function. 06:08.000 --> 06:13.640 This element in the domain negative two is paired only with ten. 06:13.680 --> 06:22.182 The element zero, however, is paired both with five and negative eight. Since this element in 06:22.240 --> 06:41.001 the domain is paired with more than one element in the range, this relation is not a function. 06:41.040 --> 06:48.575 Relations and functions can be described by the graph of the relation. 06:48.640 --> 06:54.581 This picture shows a relation because it has four ordered pairs identified, and recall that 06:54.640 --> 07:03.156 a relation is a set of ordered pairs. Is the graph shown the graph of a function? 07:03.200 --> 07:12.266 Let's look a little more closely at these ordered pairs. We'll write down the ordered pairs in this relation. 07:12.320 --> 07:25.379 The first ordered pair on the left, far left, side has the ordered pair negative three, negative two. 07:25.440 --> 07:46.833 This ordered pair is negative two, positive two. This ordered pair is negative two, negative three. And this 07:46.867 --> 08:02.349 final ordered pair is one, one. Recall that to be a function each x-value must be paired with exactly one y-value. 08:02.400 --> 08:09.556 Let's look at the domain of this relation. It includes the numbers negative three, 08:09.600 --> 08:20.434 negative two, and one. Notice that negative two is paired with both two and negative three. 08:20.480 --> 08:33.313 Because it is paired with more than one y-value, this is not a function. 08:33.360 --> 08:40.821 Let's also look at the graph to see what we can tell about this relation and not being a function 08:40.880 --> 08:53.300 as we look at the ordered pairs here. These two ordered pairs both share the same x-value. The x-coordinate 08:53.333 --> 09:11.084 is negative two. It is paired with both positive two and negative three. Thus again, this graph this relation is not a function. 09:11.120 --> 09:17.557 You may have noticed in the last example that when a vertical line can be drawn on the graph and it 09:17.600 --> 09:25.165 intersects the graph at more than one point, we know then that the relation is not a function 09:25.200 --> 09:31.805 because the ordered pairs that fall on that vertical line have the same x-value 09:31.840 --> 09:39.379 that are paired with more than one y-value. We can use this trait to learn about whether or 09:39.440 --> 09:47.621 not graphs are functions. This is called the Vertical Line Test. If a vertical line can be 09:47.680 --> 09:56.897 drawn so that it intersects a graph more than once, the graph is not the graph of a function. 09:56.960 --> 10:07.641 Let's take a look at this example. If I draw a vertical line through this graph 10:07.680 --> 10:15.615 anywhere on this graph, notice this vertical line will only intersect the graph 10:15.680 --> 10:24.357 one time. No matter where I draw this vertical line throughout this graph, the vertical line 10:24.400 --> 10:35.635 is only intersecting one time. Thus, every x-value here is paired with only one y-value. 10:35.680 --> 10:45.712 In this picture, if I draw a vertical line, here sure it only intersects one time with the graph. 10:45.760 --> 10:54.120 However, if I draw that vertical line here, notice that this vertical line 10:54.160 --> 11:02.996 intersects the graph more than one time. If a vertical line drawn anywhere on this graph 11:03.040 --> 11:11.238 intersects with this graph at more than one location, then the graph is not a function. 11:11.280 --> 11:21.548 So this graph is not the graph of a function. Whereas, the first graph, 11:21.600 --> 11:36.263 this one, where the vertical lines intersected only once, this is the graph of a function. 11:36.320 --> 11:42.736 Recall that the graph of a linear equation is a line. A line that is not 11:42.800 --> 11:51.945 vertical will always pass the Vertical Line Test. Thus, a non-vertical line is a function. 11:52.000 --> 12:00.353 An equation that is written in the form x equals c is a vertical line. And vertical lines 12:00.400 --> 12:06.259 clearly will not pass the Vertical Line Test, as all of the ordered pairs on that line share 12:06.320 --> 12:14.501 the same x-coordinate. Thus, an equation that is written in the form x equals c is not a function. 12:14.560 --> 12:21.941 Let's take a look at these two examples. Determine whether the equation describes a function. 12:22.000 --> 12:31.384 Our first equation is y minus five x equals three. We can recognize this as a linear equation because 12:31.440 --> 12:41.695 it can be written in the form A x plus B y equals C. In this equation, both A and B are not zero. It 12:41.760 --> 13:02.582 cannot be written in the form x equals c. It's not a vertical line. Thus, it is a function. 13:02.640 --> 13:10.523 In our second equation, we have x equals two. We should recognize this is in the form x equals c, 13:10.560 --> 13:18.498 which represents a vertical line. And again, vertical lines do not pass the Vertical Line Test. 13:18.560 --> 13:24.437 They have an infinite number of ordered pairs that have the same x-value. That x-value is 13:24.480 --> 13:44.591 paired to more than one y-value. Thus, this equation does not represent a function. 13:44.640 --> 13:51.564 When we first spoke of functions, we described a function as a relation such that every 13:51.600 --> 14:00.840 x-value in the relation corresponds to exactly one y-value in the relation. When that happens, that 14:00.880 --> 14:09.482 relation is a function. We also identified that linear equations that are not vertical represent 14:09.520 --> 14:20.126 functions because every x-value will relate only to one y-value. Thus, we extend our understanding 14:20.160 --> 14:30.904 of a function by saying that the variable y is a function of the variable x. We can substitute 14:30.960 --> 14:42.348 any value in the domain of the function for x. That would then give us a corresponding value of y. 14:42.400 --> 14:50.590 The variable x, thus, is the independent variable because any value in the domain can be assigned 14:50.640 --> 15:04.270 to x. The variable y is the dependent variable because its value depends on the value of x. 15:04.320 --> 15:12.278 We can also examine this function with a different notation. To identify that this 15:12.320 --> 15:23.957 equation represents a function, we can write this as f of x equals two x plus one. 15:24.000 --> 15:43.376 This is known as function notation. Note that the way we say this is f of x. This is not f times x. What this 15:43.440 --> 15:53.486 means is that the value of the function, which is y, depends upon the value of x. 15:53.520 --> 16:06.366 If I write f of one, in this way, this means the value of the function when x is equal to one. 16:06.400 --> 16:15.875 I can find f of one by substituting one for the x-value in the equation. So f of one 16:15.920 --> 16:28.755 would equal two times one plus one. Two times one plus one is three. So f of one is three. 16:28.800 --> 16:36.896 This can also be written as an ordered pair for the function. The x-value is one, 16:36.960 --> 16:45.538 and the y-value is three. Common letters used to represent functions include f, 16:45.600 --> 16:55.615 g, and h. So you may see f of x, g of x, h of x. Quite frankly, many letters, 16:55.680 --> 17:03.389 any letter, can be used to represent a function. These are just the most common. 17:03.440 --> 17:12.198 Consider the function f of x equals x squared plus three. Let's find these three values beginning 17:12.240 --> 17:21.708 with f of negative four. To find f of negative four, we will substitute negative four into 17:21.760 --> 17:35.621 the x-value of the function, giving us negative four squared plus three. Negative four squared 17:35.680 --> 17:47.567 is positive sixteen, and then we are adding three, giving us nineteen. So f of negative four 17:47.600 --> 17:58.978 is nineteen. Recall, I can write this as an ordered pair as well. This notation means that the ordered pair 17:59.011 --> 18:12.358 negative four, nineteen is on the graph of this function. I do want to point out the various uses of parentheses here. 18:12.391 --> 18:22.135 Notice that here f of negative four. Remember, we talked about the fact that this does not mean f times negative four. 18:22.168 --> 18:31.310 It means the value of the function when x is negative four. This set of parentheses here means that we are 18:31.360 --> 18:40.520 multiplying negative four times itself two times. This is a multiplication because of the exponent. 18:40.560 --> 18:48.594 And then this set of parentheses is used to represent an ordered pair negative four, nineteen. 18:48.640 --> 18:55.701 They are all parentheses, but yet they have specific uses and meanings in these problems. 18:55.760 --> 19:05.545 Let's look at the next example. We're to find f of zero. We will substitute zero for the x in the 19:05.600 --> 19:18.024 function. Giving us zero squared plus three. Zero times zero is zero, and then plus three 19:18.080 --> 19:28.601 gives us three. So f of zero is equal to three. This is the value of the function when x is zero. 19:28.640 --> 19:41.080 It can be listed as an ordered pair on the graph of this function. The ordered pair is zero, three. 19:41.120 --> 19:51.390 And our last problem is f of five. Again, we will substitute five for the x in the function, 19:51.440 --> 20:01.467 giving us five squared plus three. Five squared is five times five or twenty-five, 20:01.520 --> 20:12.278 and then we are adding three, giving us twenty-eight. So f of five is twenty-eight. 20:12.320 --> 20:33.920 And again, this can be written as an ordered pair on the graph of the function five, twenty-eight.