Lesson video

Thank you for joining us for this Transformations of Graphs lesson.

My name is Ms. Davis and I'm gonna be helping you as you work your way through.

There's lots of exciting things to explore today, as well as things to recap that we've looked at before.

To make sure that you're in a nice quiet place that you can explore this and that you pause the video and have a real think each time I offer you a question.

And then I'll help you with any hints, any ideas that will help you develop your mathematics skills.

I'm really excited to explore this topic, I hope you are too.

So let's get started.

Welcome to this lesson on checking and securing understanding of function notation.

We're gonna recap everything we know about functions today, focusing on defining and using function notation.

We're gonna revisit these keywords as they come up in our lesson.

If you'd like to read through those now, pause the video and do so.

We're gonna start then by looking at function vocabulary.

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

We call the set of values that we input into a function the domain.

The set of values that these map to is called the range.

So let's take the function three x minus one.

The domain could include the values below, I've put them into a table to help us.

To get the values in the range, we can apply the function three x minus one to each value in the domain.

So if we multiply these by three and subtract one, our range is gonna include these values.

Now we can graph this function.

So let's have a look if we have the domain on the x axis, the range on the y axis, and we can plot those values.

Our domain includes non integer values, even though I picked in integer values to my table, I can have non integer values that are between those.

And our domain could also include values beyond those that we choose for our table.

I went between negative two and three for our x values, but actually our domain could include any value unless I put a restriction on it.

If we graph this function for all possible inputs, it would form this linear graph.

We call this the graph of the function three x minus one.

So we can input any value of x and a valid output.

The domain therefore is any value of x, unless we decide to restrict that domain.

But any value of x can be used as an input.

Because this is a linear function, there'll be an input value which generates any given output.

There's no values that we can't have as an output.

So the range also includes all possible values.

Quick check then, I'd like you to fill in the blanks for this function.

So our table has values in the domain and they're gonna map to values in the range.

Our function then would be two x plus three.

Looking on our x axis, the y axis is the range, the x axis is the domain.

Then this is the graph of the function, two x plus three.

This function uniquely maps values in the domain to values in the range, we're just getting used to using that language.

It's a similar skill to things we've done before when we've drawn graphs and equations of lines, we're now thinking about this as a function and using that language.

Jim has drawn the graph of y equals x squared plus one.

He says, is x squared plus one a valid function? Well, let's return to our definition, afunction uniquely maps the values of one set to values of another set.

So negative two maps to five, negative one maps to two, zero maps to one, one maps to two, and two maps to five.

So yes, each value of x only maps to one value of x squared plus one.

So this is a function.

All values of x can be a valid input, I can use x as any value.

It doesn't matter that there's different x values that map to the same y values, so negative one and one both mapped to two.

That's absolutely fine, as long as they're each only mapped to one value.

It'd be no good if negative one mapped to two, but also four.

If the domain is any value of x, what is the range of this function? Pause video, what do you think? Right, well the range is any value greater than or equal to one.

You'll see we have a minimum at zero one, and there are no values less than one on our graph.

So the range is any value greater than or equal to one.

We can write our domain and range using set notation.

The domain includes all values of x.

So you may see that written like this.

So we have the set values x where x is any value.

You may also see this where x belongs to the set greater than negative infinity, less than infinity.

The range only includes values greater than or equal to one.

So you may see it written like this, x squared plus one where x squared plus one is greater than or equal to one.

If y is defined to be the function, e.

g, y equals x squared plus one.

Then the range can be written as y belongs to the set of values greater than or equal to one.

Quick check, the domain of the function x squared plus two x minus one is the set of values x where x is any value.

If y equals x squared plus two x minus one, what is the range of this function? I've drawn this function to help you.

So the range is y where y is greater than or equal to negative two.

You can see we've got a minimum point at negative one negative two.

There's no values less than negative two.

We've looked at a linear and a quadratic function already, there are many different types of functions.

Here's the graph of the function, sin of x.

What is the range of this function? Write values between negative one and one inclusive.

We can write that as sin of x where sin of x is greater than or equal to negative one, less than or equal to one.

We can restrict the domain to only include certain values of x.

If we restrict the domain to include the set of values X where x is greater than or equal to zero, but less than or equal to 180.

What is the range of this function now? Well, let's have a look.

We've now restricted our inputs, so they can only be between zero and 180 inclusive.

So there's our graph.

The range is now sin of x where sin of x is greater than or equal to zero, less than or equal to one.

Your turn, what is the range if we restrict the domain to x where x is greater than or equal to 180, but less than or equal to 360, what do you think? If we look at that section of the graph, the range, if we restrict the domain that way will be sin of x where sin of x is greater than or equal to negative one, less than or equal to zero.

Restricting the domain allows us to make more expressions into valid functions.

So this is the graph of y equals one over x.

Y is one over x where x is any value, not a valid function.

What do you think? Right, well, x equals zero is not a valid input.

So the function does not uniquely map every value in the domain to a value in the range, 'cause x is in the domain and that doesn't map to any values in the range.

One divided by zero is undefined.

So what we can do is we can restrict the domain to the set of values x, where x is not equal to zero, and then this is now a valid function, because x equals zero is not in our domain anymore.

So every other value does map to a unique value in the range.

The range would then be written as the set of values y where y is not equal to zero.

Or we could restrict the domain a different way.

Here we've restricted the domain to values of x where x is greater than zero.

And this is now also a valid function, but with a range of y where y is greater than zero.

Quick check, I'd like you to match up these different domain restrictions with their corresponding ranges for the function sin of x.

I've drawn a section of the graph of sin of x to help you.

Let's have a look.

If we look at values of x where x is greater than zero, that's gonna have a range sin of x where sin of x is greater than or equal to negative one, less than or equal to one.

If I restrict the domain to values of x greater than or equal to negative 90, less than or equal to zero, then the range is gonna be sin of x where sin of x is greater than or equal to negative one, less than or equal to zero.

And finally, if I restrict the domain, so we're looking at values greater than or equal to 90, less than or equal to 180 for our inputs, then sin of x can be a value where sin of x is greater than or equal to zero, less than or equal to one.

Time for a practise.

I'd like you to complete the table of values for this function.

Draw the graph of the function, label the domain and the range on the graph, and then complete the sentence below.

Give that one a go.

For question two, again, I'd like you to complete the table of values for that function.

Draw the graph of the function, x squared minus four x, and then pick the correct domain and fill in the gaps for the range.

Give that one a go.

For question three a, I'd like you to draw the graph of the function five minus x squared, but with restricted domain, x where x is grade than or equal to negative three, less than or equal to three.

You can use a table of values to help you or any of your quadratic graph skills.

Then answer questions B and C.

For question four, you've got the graph of the function cos of x.

I'd like you to tell me the range of the function cos of x, and then the range if we restrict the domain.

Then to C, you got a bit of a challenge, I've told you the range of the function.

I'd like you to fill in the blank for a possible restriction for the domain.

Give those ones a go.

Let's have a look.

The more you use this language of domain and range and function, the easier you'll find these sorts of questions.

They're all skills you've done before just getting used to that new language.

So there's my table of values, and we're looking for the function three x minus two.

On your graph, the domain is labelled on the x axis, the range on the y axis.

This is a graph of the function, three x minus two.

Question two, there's your table of values, and then you should have a parabola, look for those key features.

So it crosses the x axis at zero zero and four zero, and it has a minimum at two negative four.

The domain could be any value of x, so that can be written as the set of values x where x is greater than negative infinity, less than infinity.

And then for the range, we've got all values y where y is greater than or equal to negative four.

For question three, you should have this negative parabola with a maximum at zero five.

The range then is values five minus x squared where five minus x squared is greater than or equal to negative four, less than or equal to five.

Because we restricted that domain, the smallest value we can have as an output is negative four and the largest value is five.

If we change the domain restriction as x is a value where x is greater than zero, so a positive value, then we've still got a maximum.

We still cannot be greater than five, but there's no minimum value this time.

So you can write that range as five minus x squared where five minus x squared is less than five.

And finally, the range of the function cos of x for any input of x is cos of x where cos of x is greater than or equal to negative one, less than or equal to one, same range as sin of x if that doesn't have a restricted domain.

When we restrict the domain for values between negative 90 and 90 inclusive, then our range would be cos of x, where cos of x is greater than or equal to zero, less than or equal to one.

Now if the range of a function were values between negative one and zero inclusive, and one way we could restrict the domain would be to look at the set of values x where x is greater than or equal to 180, but less than or equal to 270.

Because I told you that bottom constraint had to be 180, then the top constraints gonna have to be 270 because we don't want to include any values greater than zero, and that's where the graph crosses the x axis after x equals 180.

Just getting used to these graphs and exploring them and seeing what happens at different sections.

So now we're gonna remind ourselves of function notation.

When working with functions, we can use notation to be more efficient.

So there's our function three x minus one.

We could say this is the function, three x minus one, or we can write f of x equals three x minus one.

Once we have defined f of x, we can use f of x instead of writing the whole function, three x minus one just to be more efficient.

So we could change the labels on our graph.

If we've got x as our domain, we could have f of x as our range.

Now we read this as f of x or the function of x.

What we're doing is we're applying the given function to x.

Jacob says like sin of x is read, sin of x, and means we're applying the sin function to x.

So you would've seen this before in other areas of mathematics.

Sometimes you might see a function written like this, y equals f of x where f of x equals 0.

5 x plus three.

What that means is because we've defined f of x as y, we could use Y, we could use f of x or we could use the actual function 0.

5 x plus three, we can use those interchangeably.

So we could label our axes as y or f of x or with the actual function, 0.

5 x plus three.

Which of these statements are true? What do you think? Right, well if you picked D, that notation does not mean f multiplied by x.

It means the function applied to x.

It definitely does not mean f to the power of x, and it definitely also does not mean f lots of x, that'd be the same as f multiplied by x.

So being really clear that this notation, which you may not be really familiar with yet means the function f of x.

If we define the function, we can then apply that function to different values of x.

Using this notation allows us to write the range of a function in a more efficient way.

The domain of f of x where f of x equals x square plus two is the set of values x where x is any value.

Because we've defined our function notation, we can now write the range as a set of values f of x where f of x is greater than or equal to two.

It just means we don't have to write the whole function x square plus two, we can just write f of x.

If working with multiple functions, we can use different letters for each function.

So we've used f of x a lot, but you can also use g of x or h of x or p of x.

You'll see lots of different letters used.

So here we've got y equals f of x where f of x is three x plus five, and y equals g of x where of x equals seven minus two x.

We could graph these on the same axis if we wish.

So there's y equals f of x and y equals g of x.

Another thing to remember we can do, we can substitute a set value into a function.

We just replace x with the given value.

So if I wanted to substitute one into the function f of x, I can write that f of one.

So f of one would be three lots of one plus five, or eight, so f of one would be eight.

And we can do that with any input.

F of two is 11, f of three is 14.

And we can do the same with g.

So g of one, we just put one into our functions, seven minus two x.

G of one is five, do the same with g of two, and g of three.

Quick check then, I'd like you to match the notation with the correct value.

You'll see that I've defined f of x, g of x and h of x on the left hand side.

I want you to evaluate those for different inputs.

Give those a go.

Let's have a look.

So f of three is two lots of three plus five, which is 11.

G of three is three squared plus five, which is 14.

H of three is three lots of three minus two, just seven.

And if we do the same for the input negative one, f of negative one is 3, g of negative one is six, and h of negative one is negative five.

The inverse of a function reverses the mapping of the original function.

So we have the function f of x equals two x plus five.

Then let's pick an input.

So let's say f of negative one, that's gonna be three, two lots of negative one plus five.

Let's try another one, f of zero is gonna be five and f of one is gonna be seven.

Now the inverse of the function will map three, five, and seven back to negative one zero and one respectively.

Now the notation we use is this notation here and that denotes the inverse of f of x.

We read that as f inverse of x.

Jacob says, again, this is like the trick functions.

That notation there is read sin inverse of x and is the inverse of the function sin of x.

So what would we need to do to get from the outputs back to the inputs for this example? Pause the video.

Right, well, we need to subtract five then divide by two.

So we can write the inverse function as f inverse of x equals x minus five all over two.

And let's try that.

So if we input three, 'cause that was our output of f of negative one.

So if we input that into our inverse function, we've got three minus five all over two and that gives us negative one.

So that does take us back to the input of the original function.

Let's do the same with five and that gives us zero, and f inverse of seven gives us one.

So you can see our inverse function reverses the mapping and takes the output back to the input.

So that is the correct inverse function.

All right, which of these statements are true? Have a read, what do you think? Well then, if you said B and D.

So for A, f inverse of x is not the same as evaluating f of negative one.

They're completely different things.

That symbol that looks like a power of negative one is just the way we denote the inverse of a function.

So we read that as f inverse x, and that is not the same as f to the power of negative one, which is written as one over f.

It's just the notation we are using.

So do not get that confused with your work on negative indices.

Read it out loud as f inverse of x.

And f inverse x maps values from the range of the original function back to the domain.

We saw that with our previous example.

Time for practise.

I'd like you to evaluate the following.

You've got three functions to find at the top.

E to evaluate those for different inputs.

Give those a go.

For question two, you've got four functions.

I've given you the function, the domain, and for most of them, I've also given you the graphs.

I'd like you to match those to the correct range.

You may want to start by drawing the graph for A and then you can match up those ranges, off you go.

For question three, I'd like you to use the partially filled in table of values to draw the graphs of y equals f of x and y equals g of x.

So you may want to start by filling in the table of values, then drawing those on the axis.

Don't forget to label which one is which.

And then tell me what the range is of each function.

For question four, if h of x equals three x plus six, which of these is the correct notation for the inverse function? Then I'd like you to evaluate h of two and h of negative two and then substitute your values into the inverse function that you chose in part A to check it is correct.

When you're happy with those, come back and we'll look at the answers.

Let's have a look.

So g of one is gonna be three, h of zero is one, f of three is four.

H of one is one, g of negative two, be careful with your priority of operations here.

You'd square your negative two first, multiply it by two, and then add negative two.

You get six.

F of negative seven, you get negative 36.

F of 2.

2, you get 0.

8.

G of negative 0.

5 gives you zero, and h of two thirds, well, two thirds cubed is eight over 27, subtract two thirds plus one, I've put them over a common denominator to help me and he gets 17 over 27.

Pause the video if you want to look at any of my work here.

For two, so you can see that I've sketched the graph of f of x equals x squared.

The range of that one is the set of values f of x where f of x is greater than or equal to zero.

For B, the range is the set of values f of x where f of x is greater than or equal to negative one, less than or equal to one.

For C, you'll see I've restricted the domain to values of x where x is greater than zero.

So the range is f of x where f of x is greater than zero.

Zero cannot be included, there's an asymptote at y equals zero.

And finally, we've got the range as the set of values f of x where f of x is less than or equal to zero.

Question three, I filled in the rest of my tables and then I've sketched those two graphs.

Don't forget to label them f of x and g of x.

The range of the functions then.

So for f of x, the range is the set of values f of x where f of x is greater than or equal to negative four.

Whereas for g of x, the range is the set of values g of x where g of x is less than or equal to six.

And finally the correct notation is that third one, h inverse of x is x minus six all over three.

So if we substitute two into our original function, we get 12, we substitute negative two into our original function, we get zero.

So if we substitute 12 and zero into our inverse function, they should map back to two and negative two.

Let's have a look.

H inverse of 12 is 12 minus six over three, which is two.

H inverse of zero, zero minus six over three, which is negative two.

So our inverse function maps the outputs back to the corresponding inputs.

So it is correct.

Well done, so we have recapped all those things that we have learned previously about functions.

If you want to go into those in more detail, you might wanna load some lessons on the functions topic.

So we've reminded ourselves of what a function is and this notation of f of x and the fact that it means a function of x, don't forget that other letters can be used as well.

Thank you for choosing to learn with us today.

Look forward to seeing you back for another lesson.

Maybe we can put these into practise and have a look at what else we can do with functions.

Have a great rest of your day.