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.