WEBVTT
00:00:01.410 --> 00:00:03.910
Let’s explore how we extract function rules.
00:00:05.550 --> 00:00:12.140
A function is a relationship that assigns exactly one output value for each input value.
00:00:14.130 --> 00:00:16.220
Let’s take a closer look at what that means.
00:00:17.750 --> 00:00:19.950
Let’s take the example from the last slide.
00:00:21.860 --> 00:00:30.870
We input a two, something happens to it inside the machine, and a four comes out on the other side.
00:00:33.140 --> 00:00:38.390
We input a three and out comes a six.
00:00:40.990 --> 00:00:43.700
In goes a four and out comes an eight.
00:00:46.140 --> 00:00:50.000
What’s happening on the inside is called the function rule.
00:00:52.550 --> 00:00:55.980
Before we move forward though, there’s some other words you need to know.
00:00:58.010 --> 00:01:02.290
We use the word input to talk about what we start with in the function.
00:01:03.470 --> 00:01:06.030
But we also call that the 𝑥-value.
00:01:07.600 --> 00:01:09.650
We could call it simply 𝑥.
00:01:10.820 --> 00:01:13.770
It’s also called the domain of a function.
00:01:14.140 --> 00:01:27.900
So all four of these words are speaking about the same thing: the input, the 𝑥-value, 𝑥, and the domain, what value we’re starting with on the function.
00:01:29.390 --> 00:01:32.710
Now you’re wondering, “Does the output have other names?”
00:01:34.100 --> 00:01:43.490
And yes, it does: the 𝑦-value, 𝑦, or the range.
00:01:46.060 --> 00:01:48.700
Okay, back to our original example.
00:01:50.640 --> 00:01:55.110
Let’s take the data that we were given and turn it into a function table.
00:01:56.540 --> 00:02:03.660
Remember that our input will be our 𝑥-value and our output belongs in the 𝑦-value.
00:02:05.460 --> 00:02:07.600
Our function table would look like this.
00:02:09.590 --> 00:02:15.040
Now, we wanna try and answer the question “what is the function rule for this table?”
00:02:17.310 --> 00:02:20.590
What happens to our two to produce a four?
00:02:22.150 --> 00:02:28.870
You could say two plus two is four-two plus two is four.
00:02:30.590 --> 00:02:35.330
Our function rule has to work for every 𝑥 and 𝑦 in the table.
00:02:36.770 --> 00:02:38.500
Let’s check and see if it does.
00:02:40.160 --> 00:02:42.700
Three plus two is five.
00:02:45.010 --> 00:02:49.060
But in our table, the output from three is six.
00:02:50.470 --> 00:02:53.760
This means that our function rule is not plus two.
00:02:54.190 --> 00:02:55.800
We need to think of something else.
00:02:58.040 --> 00:02:59.520
We need another operation.
00:03:01.410 --> 00:03:03.890
What about two times two?
00:03:06.000 --> 00:03:20.620
Two times two gives us four, three times two gives us six, four times two gives us eight, and five times two gives us 10.
00:03:22.470 --> 00:03:29.590
Our input or our 𝑥-value multiplied by two equals our output — our 𝑦-value.
00:03:30.840 --> 00:03:33.660
Two 𝑥 is our function rule.
00:03:36.800 --> 00:03:41.450
Let’s look at this question: Is the following relationship a function?
00:03:43.330 --> 00:03:46.650
We need to remember the definition of a function.
00:03:48.080 --> 00:04:00.860
A function is a relationship that assigns exactly one output for every input — exactly one output for each input.
00:04:02.380 --> 00:04:09.790
Let’s use a function table to see if this relationship assigns exactly one output for each input.
00:04:11.380 --> 00:04:15.390
When we put in four, seven is the output.
00:04:17.360 --> 00:04:20.890
When we input five, the output is two.
00:04:22.040 --> 00:04:25.020
The problem is that’s not the only output.
00:04:26.830 --> 00:04:28.130
We can stop right here.
00:04:29.610 --> 00:04:35.110
This relationship has assigned two values as the output for five.
00:04:36.090 --> 00:04:39.790
And therefore, we cannot call the relationship a function.
00:04:41.290 --> 00:04:45.810
The answer to the question “Is this relationship a function?” is no.
00:04:47.090 --> 00:04:51.420
We know that’s true because of the definition of a function.
00:04:52.990 --> 00:04:57.460
Here’s another example, where we need to find the rule for a function.
00:04:59.210 --> 00:05:03.090
We’re given a table and asked to find the rule.
00:05:05.010 --> 00:05:08.910
We need to figure out what happens inside our machine.
00:05:09.520 --> 00:05:15.990
What do we do to our 𝑥-values, our input, that will give us these outputs every time?
00:05:17.480 --> 00:05:24.160
If we look at the outputs we’re given, we can see that from 10 to 14, we’ve added four.
00:05:25.050 --> 00:05:28.100
And from 14 to 18, we’ve added four.
00:05:29.460 --> 00:05:32.970
In fact, 18 plus four is 22.
00:05:33.330 --> 00:05:35.540
22 plus four is 26.
00:05:36.780 --> 00:05:39.760
This is our first clue into what’s happening here.
00:05:41.160 --> 00:05:46.460
Next, we wanna ask, “What would be the output if the input was zero?”
00:05:47.830 --> 00:05:51.000
For all of our other outputs, we’ve been adding four.
00:05:52.650 --> 00:06:01.390
If we subtract four from 10, we can figure out what the function would be at zero, which is six.
00:06:04.040 --> 00:06:06.220
This is going to be really helpful for us.
00:06:07.800 --> 00:06:11.970
What operation can take zero and give us six?
00:06:14.760 --> 00:06:15.930
Plus six right?
00:06:17.470 --> 00:06:22.600
That would mean we take our 𝑥, we add six, and that gives us 𝑦.
00:06:24.200 --> 00:06:27.050
Well, zero plus six equals 𝑦.
00:06:28.830 --> 00:06:32.830
Now, does one plus six equal 10?
00:06:34.470 --> 00:06:35.400
It doesn’t work.
00:06:35.740 --> 00:06:36.920
So we have a problem.
00:06:38.580 --> 00:06:40.120
Something is wrong here.
00:06:41.190 --> 00:06:44.400
What plus six equals 10?
00:06:45.880 --> 00:06:47.030
Four of course.
00:06:48.640 --> 00:06:51.680
But our 𝑥-value is one and not four.
00:06:53.270 --> 00:07:01.300
How about this: what if we turned our 𝑥-value one into four by multiplying 𝑥 by four?
00:07:02.820 --> 00:07:06.280
Four times one plus six equals 10.
00:07:07.670 --> 00:07:14.740
Let’s go back and check our zero: zero times four plus six equals six.
00:07:16.510 --> 00:07:23.390
Testing 𝑥-value of two, two times four is eight plus six is 14.
00:07:26.360 --> 00:07:32.770
It’s true for three and 18, true for four and 22.
00:07:35.160 --> 00:07:40.850
And finally, five times four is 20 plus six is 26.
00:07:42.380 --> 00:07:48.040
We have our function rule: the function rule is four 𝑥 plus six.
00:07:49.050 --> 00:07:55.910
The format we use for writing functions is 𝑦 equals whatever your function rule is.
00:07:56.320 --> 00:08:03.410
In our case, we have 𝑦 equals four 𝑥 plus six because that is the function rule for this table.
00:08:05.670 --> 00:08:09.610
Let’s take a minute and look at these two words: domain and range.
00:08:10.960 --> 00:08:19.240
Domain is the set of all the input values and range is the set of all the output values.
00:08:20.940 --> 00:08:21.900
What does that mean?
00:08:23.570 --> 00:08:29.300
It looks something like this and like this.
00:08:32.190 --> 00:08:36.970
We would say that two is part of the domain, but it’s not the whole domain.
00:08:38.610 --> 00:08:44.080
18 is part of the range or within the range, but it’s not the whole range.
00:08:45.390 --> 00:08:52.210
The domain is all of the input values and the range is all of the output values.
00:08:54.000 --> 00:08:57.600
These are the tools you need to go and solve your own function rules.