### 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.