### Video Transcript

In this video, we’re gonna learn about the terms function, domain, range, and
tables of values, or function table. We’ll also use a relation diagram. Functions are used in
lots of different strands of maths and science. And it’s really important to learn and
understand this topic.

We’re gonna look at an example of a situation where we’re gonna use a
function. And this video is designed as a first look, giving you an overview of all
these different ideas, and what they might mean, and how they might be used.

Let’s start by thinking of a situation where we might use a function,
temperature say. Some people use the Fahrenheit scale to quantify their temperatures, and
others use Celsius or centigrade. That gives us two different sets of numbers to represent
each individual temperature. For example, the freezing point of water is zero degrees on the
Celsius scale, but exactly the same temperature is called thirty-two degrees on
the Fahrenheit scale. The boiling point of water is a hundred degrees on the
Celsius scale, or two hundred and twelve degrees on the Fahrenheit scale.

Now there is a relationship between the Celsius temperature and it’s
corresponding Fahrenheit temperature. Sometimes we call that a relation. In fact, there’s a
relation in the other direction as well. So from Celsius to Fahrenheit, and from Fahrenheit to
Celsius. Each Celsius temperature relates to a specific Fahrenheit temperature. And we
call a Celsius temperature and it’s corresponding Fahrenheit temperature, an ordered pair. So
zero and thirty-two are an ordered pair and a hundred
and two hundred and twelve are an ordered pair. So we’ve given two examples there. But actually, each temperature in Celsius
relates to exactly one Fahrenheit equivalent. There aren’t any temperatures which have one
value in Celsius but two or more different possible values in Fahrenheit. So we call this a
one-to-one relation, or relationship.

So here is an example of a relation between Celsius temperatures and their
Fahrenheit equivalents. I’ve taken six temperatures and I’ve got the input values, and their corresponding output values. And when I put an input with its corresponding output, they become an ordered
pair. So for example, twenty degrees Celsius corresponds to
sixty-eight degrees Fahrenheit, so that makes an ordered pair. So this relation, or relationship, between Celsius temperatures and their
corresponding Fahrenheit temperatures is quite a special one, because it’s a one-to-one
mapping. For every Celsius temperature, there is exactly one Fahrenheit temperature. And the fact that we’ve got this special relation going on here, so for every
input value that you put in, there is only one possible output value, you know what the answer
is gonna be, that means we can call this relation a function. A relationship where every input
is related to exactly one output.

Imagine we make a special machine and we feed numbers into that machine, and
it carries out some mathematical processing. If we feed in a temperature from the Celsius
scale, it does a bit of jigging and warring. It does the processing and spits out the related
temperature from the Fahrenheit scale. Whenever we feed in a number, some magic happens in here and outputs one
related number, not two or three to choose from, but one, the answer, the output. If we think of all the possible numbers that we could choose from to input,
that set of numbers is called the domain. It’s easy to remember that word because domain
contains the word in, and it’s input. So each one input value relates to a corresponding output value. And that set
of output values is called the range.

Now this is where it can get a little bit complicated. There are infinitely
many different numbers that are valid temperatures in degrees Celsius, because it’s a
continuous thing. There are infinitesimally small variations of temperature that we can go
through, as things get hotter and hotter. Well probably in practice, temperature isn’t quite continuous because it’s a
measure of the energy levels of subatomic particles. But let’s not get too hung up on that for
now. But also with temperature there is in fact a minimum possible temperature,
and that’s called absolute zero, which is about minus two hundred and seventy-three
point one five degrees Celsius. And you can’t get colder than that. And there might even be a maximum possible temperature, and some people call
that absolute hot. The point is, the domain is the set of all possible temperatures in Celsius
and our function will process each input and convert it to an output in Fahrenheit. And that
set of numbers is called the range. So here’s the way to think, a domain is a set of pretty much infinite number
of temperatures in degrees Celsius, between absolute zero and absolute hot. And a range is the corresponding set of values in degrees Fahrenheit. Now what we can do is, we can cut down, we can limit our domain. So we might
say that we’re only going to consider temperatures between zero degrees Celsius
and a hundred degrees Celsius, from the freezing point of water to the boiling
point of water. So if we set our domain between zero and a hundred
for our input values, that will then restrict the range as well.

Okay. That was all pretty tricky. Now if we were to build the function
machine, what would have to go on inside this little box? Well to convert a Celsius temperature to a Fahrenheit temperature, you take
that Celsius temperature and times it by nine. Then divide that answer by
five, and then add thirty-two. And your answer, hey presto, is the
temperature in Fahrenheit. So if we call the temperature in Celsius 𝑥, and we call the
temperature in Fahrenheit 𝑦, we can come up with this formula, 𝑦 is equal to 𝑥 times nine divided by
five and then add thirty-two to the final answer. And perhaps a nicer way of writing that is, 𝑦 equals nine over five 𝑥
plus thirty-two. Now because we said that every possible input value of 𝑥 will
always lead to a specific output value of 𝑦, we know that this is a function. And
that means that we can use some alternative notation. We can replace the 𝑦 with 𝑓 of 𝑥, so 𝑓 of 𝑥
is equal to nine over five 𝑥 plus thirty-two. Now don’t worry too much about this notation for now. It’s just a way of
saying that this temperature conversion is a function in terms of 𝑥. It just
relies on an input value and that will generate a unique output value in terms of degrees
Fahrenheit.

Now let’s have a look at how to use this function in practice. Let’s say we want to convert fifteen degrees Celsius into
Fahrenheit. Wherever it said 𝑥 in our original formula, we just replace that
with fifteen because that’s the specific temperature in Celsius we’re trying to
convert. And when we do that calculation, we get an answer of fifty-nine
degrees Fahrenheit. So fifteen degrees Celsius is fifty-nine degrees
Fahrenheit.

Let’s do another example. Seventy-five degrees Celsius, what’s
that in Fahrenheit? Well this time, the number that we have to replace 𝑥 with is
seventy-five. So 𝑓 of seventy-five is nine over five times seventy-five
plus thirty-two, which is one hundred and sixty-seven degrees Fahrenheit.

Now we could feed lots more of these input values into our formula and
collect all the corresponding output values and record them all in a table. So I’m just gonna do this with a small number of values, but you could do as
many, or as few, as you like. So we’re just taking zero, twenty,
forty, sixty, eighty, and a hundred
degrees Celsius. And then, we replace 𝑥 with each of those numbers in turn in our
formula and calculate the results. And these are the Fahrenheit temperatures that we get. So this table is
called a table of values, or a function table. And remember, the set of values that we put into the function are called the
domain. And the corresponding set of results we get out from the function are called
the range. Now we said in this example that we were limiting the domain that we were
looking at to zero to a hundred degrees Celsius. But in fact, the
domain would also consist of all of those infinite number of possible temperatures between
zero and a hundred as well. So the numbers we’ve got listed here
is not the complete domain.

Now I’m gonna take our set of ordered pairs, our inputs and their
corresponding outputs, and I’m gonna use those as coordinates and plot a graph of these
values. So when we plot in the graph, we always use the input values as our
𝑥-coordinates and our output values as our 𝑦-coordinates. So first
of all, we can do zero, thirty-two, that’s about there. And twenty, sixty-eight, that’s about there. And forty, a hundred and four, and so on. Now we can see from the graph that there is a linear pattern. And although we
only plotted a small number of points, six points, we can draw the line that will graph all of
the infinite number of input and output pairs in between them. And we can use that graph to
look up the values. So for example, sixty degrees Celsius, we could go up here and
that maps to a hundred and forty degrees Fahrenheit. Or, seventy-two degrees Celsius maps to a hundred and
sixty degrees Fahrenheit. Well there we’ll be able to see the limitations of this
graph because that’s only approximate, because we can’t draw this a hundred percent
accurately. If we’re able to do the math, seventy-two degrees Celsius actually
comes out to be a hundred and sixty-one point six degrees Fahrenheit. So because of the limitations of my drawing skills, and the-the accuracy of
the graph, we can see that this is only an approximate method. But it’s a really great visual
representation of the relationship between temperature in Celsius and temperature in
Fahrenheit. We can see that it makes a straight line representation.

So let’s just summarize the journey we’ve been on then. First, we looked at
temperatures and we created a function box which would convert Celsius values into their
corresponding Fahrenheit values. Then we called our inputs 𝑥, and we used what was going on in
the magic function box to create ourselves a function, using function notation. Then we said that the domain was the set of valid input values for numbers we
could plug in to our function. And we said that the range was the set of the corresponding output values, or
the results. And when we drew the relation, we saw that each input maps to exactly one
output. So when we put something in, we know we’re only gonna get exactly one answer out.
We’re not gonna have to make a choice, as to, is it gonna be this answer or that answer. That
feature means that we can call this relation a function. Then we created a table of values, or a function table, by generating a
specific set of inputs putting them into the function and getting our specific set of
corresponding outputs.

And lastly, each input along with its corresponding output generates an
ordered pair, which we can use as coordinates to produce a graph. On a graph we always use the input as the 𝑥-coordinate and the
output as the 𝑦-coordinate. And looking at the shape of the graph, we can understand a bit more about the
relationship between inputs and outputs. And in this case, that was a linear relationship. And this means with temperatures, no matter how hot or cold it is, if I
increase the temperature by one degrees Celsius, I will always get the same increase in temperature in Fahrenheit, about
one point eight degrees in this case. So the straight line is telling us that
no matter how hot or cold it is, an increase of one degree Celsius will generate
the same increase in temperature in Fahrenheit.

Now we’ve got lots of other videos looking at each of those sections in more
detail, but hopefully that’s given you a kind of overview of how all these things fit together
and why we use them.