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 ten. 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 the 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 ten to fourteen we’ve added four and from fourteen to eighteen we’ve added four. In fact eighteen plus four is twenty-two; twenty-two plus four is twenty-six. 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 ten, 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 ten? It doesn’t work, so we have a problem. Something is wrong here. What plus six equals ten? 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 ten. 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 fourteen. It’s true for three and eighteen, true for four and twenty-two. And finally five times four is twenty plus six is twenty-six. 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. Eighteen 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 solve your own function rules.