Video: Extracting Function Rules

Introduce functions through values in function tables and learn that functions have exactly one output for each input. Find, write, and solve function rules. Learn about the definitions of domain (the set of inputs) and range (the set of outputs).

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Video Transcript

Let’s explore how we extract function rules. A function is a relationship that assigns exactly one output value for each input value. Let’s take a closer look at what that means.

Let’s take the example from the last slide. We input a two, something happens to it inside the machine, and a four comes out on the other side. We input a three and out comes a six. In goes a four and out comes an eight. What’s happening on the inside is called the function rule. Before we move forward though, there’s some other words you need to know. We use the word input to talk about what we start with in the function. But we also call that the 𝑥-value. We could call it simply 𝑥. It’s also called the domain of a function. 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. Now you’re wondering, “Does the output have other names?” And yes, it does: the 𝑦-value, 𝑦, or the range.

Okay, back to our original example.

Let’s take the data that we were given and turn it into a function table.

Remember that our input will be our 𝑥-value and our output belongs in the 𝑦-value. Our function table would look like this. Now, we wanna try and answer the question “what is the function rule for this table?” What happens to our two to produce a four? You could say two plus two is four-two plus two is four. Our function rule has to work for every 𝑥 and 𝑦 in the table. Let’s check and see if it does. Three plus two is five. But in our table, the output from three is six. This means that our function rule is not plus two. We need to think of something else. We need another operation. What about two times two? 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. Our input or our 𝑥-value multiplied by two equals our output — our 𝑦-value. Two 𝑥 is our function rule.

Let’s look at this question: Is the following relationship a function?

We need to remember the definition of a function. A function is a relationship that assigns exactly one output for every input — exactly one output for each input. Let’s use a function table to see if this relationship assigns exactly one output for each input. When we put in four, seven is the output. When we input five, the output is two. The problem is that’s not the only output. We can stop right here. This relationship has assigned two values as the output for five. And therefore, we cannot call the relationship a function. The answer to the question “Is this relationship a function?” is no. We know that’s true because of the definition of a function.

Here’s another example, where we need to find the rule for a function. We’re given a table and asked to find the rule. We need to figure out what happens inside our machine. What do we do to our 𝑥-values, our input, that will give us these outputs every time?

If we look at the outputs we’re given, we can see that from 10 to 14, we’ve added four. And from 14 to 18, we’ve added four. In fact, 18 plus four is 22. 22 plus four is 26. This is our first clue into what’s happening here. Next, we wanna ask, “What would be the output if the input was zero?” For all of our other outputs, we’ve been adding four. If we subtract four from 10, we can figure out what the function would be at zero, which is six. This is going to be really helpful for us. What operation can take zero and give us six? Plus six right? That would mean we take our 𝑥, we add six, and that gives us 𝑦. Well, zero plus six equals 𝑦. Now, does one plus six equal 10? It doesn’t work. So we have a problem. Something is wrong here. What plus six equals 10? Four of course. But our 𝑥-value is one and not four.

How about this: what if we turned our 𝑥-value one into four by multiplying 𝑥 by four? Four times one plus six equals 10. Let’s go back and check our zero: zero times four plus six equals six. Testing 𝑥-value of two, two times four is eight plus six is 14. It’s true for three and 18, true for four and 22. And finally, five times four is 20 plus six is 26. We have our function rule: the function rule is four 𝑥 plus six.

The format we use for writing functions is 𝑦 equals whatever your function rule is. In our case, we have 𝑦 equals four 𝑥 plus six because that is the function rule for this table. Let’s take a minute and look at these two words: domain and range.

Domain is the set of all the input values and range is the set of all the output values. What does that mean? It looks something like this and like this. We would say that two is part of the domain, but it’s not the whole domain. 18 is part of the range or within the range, but it’s not the whole range. The domain is all of the input values and the range is all of the output values. These are the tools you need to go and solve your own function rules.

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