Lesson video

Hello.

Mr. Robson here.

Welcome to Maths.

Good decision by you to be here.

Today, we're checking on securing our understanding of functions.

It's no coincidence that you can't spell functions without the word fun, so let's get started.

Our learning outcome is that we'll be able to appreciate that a function is a mathematical relationship that uniquely maps a set of numbers onto another set of numbers.

Some keywords we're gonna hear today, function.

A function is a mathematical relationship that uniquely maps values of one set to the values of another set.

Domain.

The domain of a function is the set of values that the mapping is performed on.

And range.

The range of a function is the set of values mapped to by the function and the stated domain.

Look out for all those keywords throughout our lesson today.

Two parts to our learning, and we're gonna begin with function vocabulary.

A function is a mathematical relationship that uniquely maps values of one set to the values of another set.

A simple function like times 2 maps these inputs onto these outputs.

The inputs 0 to 5 mapping to the outputs 0, 2, 4, 6, 8, 10.

The function times 2 uniquely maps 0 to 0 and no other value.

The same function uniquely maps 1 to 2 and maps it to no other value.

It maps 2 to 4 and no other value.

Can you see what we mean by the fact that a function uniquely maps? We can graph the function times 2.

We'll take our table of values and we'll plot these inputs versus outputs, with inputs being the horizontal axis and outputs being the vertical axis.

We know there are infinitely many more values between and beyond each of these integer inputs.

If we mapped all of those values, our graph would form a line.

You'll see that I've added another column to our table.

I've got a key question for you now.

What if x went through our function? Well done.

The output would indeed be 2x.

You'll have seen this called the graph of y equals 2x before.

We can also say that this is the graph of the function 2x.

When graphing functions, we don't use the words inputs and outputs.

We call our x values the domain, and we call our y values the range.

I'll reflect those two words on our axis now.

The domain is horizontal axis and the range is the vertical axis.

Quick check, you've got all that.

I'd like to fill in the blanks for this function.

There's four blank spaces on the screen here.

Can you write down what's gonna go in each of them? Pause and do that now.

Welcome back.

Let's see how we did.

Hopefully in the table there underneath domain, you wrote range.

Our domain is mapped onto our range.

Our range is labelled on the vertical axis.

We should have labelled the horizontal axis with the word domain.

If x is the input into the function, the output is 3x minus 1.

Therefore, we can say this is the graph of the function 3x minus 1.

Sofia has graphed the function x squared.

She says, "You can square negative numbers, so the range of this function includes -3, -2, and -1." Do you agree with Sofia's statement? Pause and have a think.

Welcome back.

I wonder what you said.

Sofia's realised something.

She says, "I had the labels the wrong way round.

<v ->3, -2, and -1 are in the domain</v> of this function, but not the range." We can define the range of the function x squared as greater than or equal to zero.

Can you see that in the graph? The function x squared will have no values below zero.

Quick check, you've got that.

I'd like you to complete these statements for the function 1 minus x squared.

There's five blank spaces there and some important words or some important wording needs to go in those.

Pause and populate them now.

Welcome back.

Hopefully you populated this like so.

2 and 3 are in the domain of the function 1 minus x squared, but not the range.

<v ->3 is in both the domain and range of the function.

</v> We can define the range of this function as less than or equal to 1.

You won't find any values outside of that in the range of this function.

Practise time now.

Question one, part a, I'd like you to complete the table of values.

This is the function times 5 minus 4.

Once you populate the table of values, you're onto to part b where you'll draw the graph of the function.

And then for part c, you're going to complete that sentence.

Pause and do those three things now.

For question two, you're gonna complete the table of values for the function half x plus 4.

Then for part b, once your table of values is populated, you can draw the graph of the function.

Pause and do this now.

Question three.

You're gonna complete the table of values and graph the function x squared minus 4x.

Slightly trickier function now.

I'm sure we've got this.

Once you've drawn the graph, I'd like you to categorise these values for part b.

There's seven values there.

In which category do they fall? Are they in the domain only, in the range only, or are some of them in both the domain and the range? Pause and do those things now.

Feedback time now.

Let's see how we got on.

Question one, part a, I asked you to complete the table of values.

Your table should have looked like so.

Once you've done that, you can graph the function and your graph should have looked like that.

For part c, you should have written, this is the graph of the function 5x minus 4.

Pause and check that your table of values, your graph are exactly like mine.

Question two, complete the table of values for the function half x plus 4.

That should have looked like so.

And your graph should have looked like so.

Pause and check that yours matches mine.

Question three, we're completing the table of values and graphing the function x squared minus 4x.

Table of values should have looked like so, and your graph should have looked like so.

Part b, I asked you to categorise those values.

You'll note that I've labelled my axis with domain and range.

This helps me to think about this problem.

It helps me to see that the value zero is in both the domain and the range, as is -3 and -4.

<v ->5, however, is in the domain only.

</v> 5 is in both the domain and the range.

1,000, whilst we can't see it on the graph, we can imagine it and it will be in both the domain and the range.

<v ->10 is in the domain only.

</v> For -5 and -10, wen we say they're in the domain only, it might help you to think that they will be a valid input but never an output.

Onto the second half of our learning now.

We're going to look at identifying functions.

It's important we can identify what is and what is not a function.

Here are the graphs of y equals x and y equals 1/x.

A problem arises when we graph the functions of x and 1/x.

The function x uniquely maps every possible x value to a y value.

That makes that a valid function.

This is not true for the function 1/x.

x equals 0 is not a valid input.

The function 1/x does not uniquely map every possible x value to a y value.

Therefore, it's not a valid function.

In order to be a function, every value in the domain must be able to be mapped to a value in the range.

We can make 1/x a valid function by defining what is not a valid input.

We do this by restricting the domain.

Can you see what's different now? I'm saying graph the function 1/x by adding, importantly, x can be any value except 0.

0 because that's the thing we can't input into 1/x.

Dividing by 0 is undefined, remember? So x can be any value except 0.

This is a restriction on the domain and it defines our invalid inputs.

By placing this restriction on the domain, we now have a valid function.

Quick check, you've got that.

For which of the below would we need to restrict the domain in order to make them valid functions? For a, x plus 3 over 2.

Is that valid for all inputs of x? For b, 3/2x.

Is there any x value we can't input there? For c, 3 minus 2x.

And for d, 2 over x plus 3.

Which of those will require a domain restriction? Pause, have a think, have a conversation with the person next to you.

I'll see you in a moment for the answer.

Welcome back.

Hopefully you said b would require a domain restriction and d would require a domain restriction.

For part a, x plus 3 over 2 is already valid for all values of x.

There's no x value that you can't input into that function.

The same's true for c, 3 minus 2x.

There's no x value that we can't input into that function.

But for b, it requires a domain restriction x can be any value except 0.

3 over 2 lots of 0, try typing that into your calculator and see where it gets you.

Not very far.

For D, slightly different domain restriction but a domain restriction nonetheless.

We'd have to say x can be any value except -3.

Remember, if x was -3, that denominator will be 0 and dividing by 0 is undefined.

Next check, I'd like you to match the below functions to the restricted domain that makes them valid.

Five functions, five restrictions, pause and pair those up.

Welcome back.

Hopefully you paired 5 over x minus 3 to x can be any value except 3.

The next one, 5 over x plus 3 multiplied by x minus 7, should pair that to x can be any value except -3 or 7.

For the third function, 5 over 3 lots of x minus 7, that should have been paired with x can be any value except 7.

For the fourth function, we should pair that to x can be any value except -7.

And that leaves us with the last pair, 5 over x plus 3 pairing with x can be any value except -3.

There are different types of functions.

In the case of the function 2x, we could extend our inputs infinitely in both the positive and negative direction and each value can be uniquely mapped to one output, can be uniquely mapped to one output, One input uniquely maps to one output.

I'm repeating the word one a lot for this reason.

That makes 2x a one-to-one function.

One input uniquely maps to one output.

It's a one-to-one function.

In the case of the function x squared, something different is true.

You can see it in the table of values and you can see it in the graph of x squared too.

Many inputs uniquely map to the same one output.

In the case of -1 and 1 as inputs, that's many inputs mapping to the same one output.

The same thing for -2 and positive 2 as inputs.

Many inputs mapping to the same one output.

For this reason, we call x squared a many-to-one function, many inputs mapping to the same one output.

Quick check, you've got that.

I'd like you to categorise these four functions as one-to-one or many-to-one.

Pause, have a think.

Welcome back.

Hopefully you said 3 minus x is a one-to-one function.

One input uniquely mapping to one output in each and every case.

3 minus 2x squared, that's a many-to-one function.

You can see in the table of values as well as in the graph, -2 and positive 2 has many inputs mapping into the same one output of -5.

That makes it a many-to-one function.

For the function x cubed, that's a one-to-one function also.

Whereas x cubed minus 2x squared minus x plus 2 is a many-to-one function.

In the inputs of -1, 1, and 2, That's many inputs mapping to the same one output of 0.

We've seen one-to-one functions and many-to-one functions.

Do you think a function exists which could be one-to-many? Pause, have a think about that.

Have a conversation with the person next to you.

Don't be afraid to speculate.

Do you think this could exist, a one-to-many function? Pause, I'll see you in a moment.

Welcome back.

I wonder what you thought.

That would be one input mapping to many outputs, a one-to-many function.

Root x is a one-to-one function when x is greater than or equal to 0.

One input uniquely maps to one output in the case of that function.

Positive or negative root x is different.

When I put the same inputs into my table of values, 0, 1, 2, 3, 4, I take the positive and negative roots, you can see I've got multiple outputs for the value 1, 2, 3, and 4.

The input of four maps to both positive 2 and -2.

That's one input mapping to many outputs.

Positive negative root x is an example of a one-to-many relationship.

One input maps to many non-unique outputs.

That's an important sentence.

One input maps to many non-unique outputs, non-unique.

When we input 4, the output is not unique.

We could have positive 2 or -2.

One-to-many relationships are not valid functions.

Why? Excellent question, thank you.

The reason why is because by definition, a function is a mathematical relationship that uniquely maps values of one set, the domain, to the values of another set, the range.

The clue is in the word uniquely.

When an input is 4, we don't uniquely map to positive 2 or to -2.

We're mapping to both.

That makes this an invalid function.

One-to-many functions are not valid functions.

Quick check, you've got that.

Which of these relationships form valid functions? One-to-one relationships, many-to-one relationships, or one-to-many relationships.

Pause and decide now.

Welcome back.

One-to-one, absolutely they form a valid function.

One input uniquely mapping to one output.

Many-to-one, absolutely a valid function.

Many values in the domain uniquely mapping to the same one output, that is fine.

There's a problem with the final one.

They're one-to-many relationships, not a valid function.

Each input does not uniquely map to one output, which means that this will not be a valid function.

Practise time now.

Question one, I'd like you to determine which of the following are functions.

For those that are not functions, explain why they are not.

Pause and do that now.

Question two, I'd like you to categorise the below into one-to-one and many-to-one functions.

My hint, a sketch of the graph might be useful.

It's not necessary entirely, but you might find it useful to see which ones are one-to-one and which ones are many-to-one.

Pause, have a go at these now.

Welcome back.

Feedback time.

Question one, we were determining which of the following are functions.

And for those that are not functions, we were explaining why not.

For a, absolutely, definitely a function.

It's a valid many-to-one function, no domain restriction would be required.

For b, absolutely a valid function.

It's a valid one-to-one function made valid by that domain restriction.

If we didn't have written there x can be any value except 7, that would render it an invalid function.

However, we have that domain restriction, so it's fine.

It's a valid function.

C, not a valid function.

Why not? Well done.

It's because a domain restriction is required in order to make all the inputs valid.

D, not a valid function because it's a one-to-many relationship.

7 plus or minus the root of x would be a one-to-many relationship.

It's not a valid function.

For question two, we're categorising the below into one-to-one and many-to-one functions.

Hopefully for a, you said that's a one-to-one function.

Each value in the domain uniquely maps to one value in the range.

For b, x cubed minus 4x squared plus 3x, sketch of the graph would've had you with a curve looking something like that.

A beautiful cubic curve which reveals it's indeed a many-to-one function.

There are three values in the domain uniquely mapping to a value of 0 in the range that makes it a many-to-one function.

For c, 3x squared is another many-to-one function.

Two values in the domain uniquely mapped to a value of three in the range.

You could see that from a sketch.

D, e, and f.

D was a one-to-one function, Each value in the domain uniquely mapping to one value in the range.

For e, a many-to-one function where we've got two values in the domain uniquely mapping to a value of 4 in the range.

And for f, the domain restriction makes it a one-to-one function.

I'm sure a few people were tricked into declaring that many-to-one function because all the other x squared ones we've seen so far have been.

Not true in this case because of that domain restriction, x is greater than or equal to 0.

That means it's a one-to-one function.

Each value in this domain uniquely maps to one value in the range.

Sadly, we're at the end of the lesson now.

Hope you've enjoyed it as much as I have.

What have we learned? We learned that a function is a mathematical relationship that uniquely maps values of one set to the values of another set.

Functions that uniquely map each value in the domain to one value in their range are called one-to-one functions.

If a function uniquely maps many values in its domain to the same value in the range, it's called a many-to-one function.

One-to-many functions do not exist because they do not uniquely map each value in their domain.

That's goodbye from me for now, but I look forward to seeing you again soon for more maths.