Lesson video

Hello, Mr. Robson here.

Welcome to Maths.

Superb choice to join me today.

We're building on composite functions in our learning today.

Well that's mathematical learning all over, isn't it? We're always building upon something we've seen before.

You're gonna see that in action today, so let's take a look.

A learning outcome is that we'll be able to expand our thinking on composite functions to consider a more efficient way to write composite functions comprised of many functions.

There's a lot going on there, but don't worry, it'll all become clear throughout the lesson.

There's lots of vocabulary in today's lesson and this keyword you might not yet know, iteration.

Iteration is the repeated application of a function or process in which the output of each iteration is used as the input for the next iteration.

We're gonna become very familiar with this word today.

Two parts to our learning.

Let's begin with an introduction to notation.

You invest 200 pounds in a savings account with an interest rate of 6% per annum and leave it in place for five years.

We can quickly and easily calculate how much this would be worth at the end of five years.

Putting that into your calculator will tell you it's worth 267 pounds and 65 pence at the end of five years.

Our 200 pounds was repeatedly multiplied to find this end value.

Whilst this formula quickly gives us our final value, what we've got at the end of five years, we might be interested in knowing the value of our savings each year.

This formula doesn't immediately give us that year-on-year information.

So we could go back in and alter the exponent, the time period, so to speak.

I could change that five to a four, but it was five fiddly button depressions each time, I had to go left, left, backspace, four, execute.

Is there a better way of doing this? The most efficient way to find the year-on-year values is to start by inputting the principle value 200 and pressing EXE.

EXE is your execute button.

If you have an inferior calculator to the Casio ClassWiz, it might be your = button, but if you're like me and you use a Casio ClassWiz, it's the EXE button.

We now have an answer of 200 in our calculator, which we can use as the next input by using the Ans button.

Ans times 1.

06 = 212.

What do you think the Ans was short for? Answer, of course.

Now that we've got the first two values and we've got that in our calculator display, just pressing EXE, the output becomes the next input and our calculation is repeated.

I'll press EXE.

That gave us an input of 212 to give us an output of 224 pounds and 72 pence.

Pressing EXE again inputs the 224 pounds and 72 pence to give me that year 3 output, and this process repeats.

What we end up with is a sequence of values which were multiplied by 1.

06 to get the next term in the sequence.

Our repeated calculation was a multiplication, so our values form a geometric sequence.

We could say it's a geometric sequence because it was a repeated multiplication, the same repeated calculation.

I wonder why I'm repeating that word.

It's because it's an example of iteration.

Iteration is the repeated application of a function or process in which the output of each iteration is used as the input for the next iteration.

In this case, the output of any given year became the input for the next year.

Hence it's an example of iteration.

The word iteration is closely linked to the word re-iterate.

When you are repeatedly arguing your point in a debate, you can say, "I will re-iterate that.

." and then repeat your point.

Iteration in maths, it's the same thing really.

It's the mathematical function repeating itself.

So here's the same scenario.

Our investment of 200 pounds growing at 6% per annum.

Given that it's an iterative process, we can write this scenario, the respective values, and the term-to-term calculation using iteration notation.

Let's start by efficiently writing the year numbers.

Year 0, we could write as y subscript 0.

You might hear that described as y subscript 0.

You might hear it called y0.

The lowercase y is used to show us we're talking about a year in this context, the subscript 0 tells us we're referring to year 0, the principle value.

Zero years have passed.

Zero iterations, zero calculations have occurred.

Year 1, we could write as y subscript 1.

Again, you'll hear this referred to as y1 at times.

Lower case y to show we're talking about a year in the context.

Subscript 1 tells us we're referring to year 1.

One year has passed.

One iteration, one calculation has occurred.

I'm sure you know by now what we're gonna write year 2 as.

Well done.

y subscript 2, y2.

In iteration notation how would we more efficiently write year 5? Pause, take your pick from those three.

Welcome back.

I do hope you said option C, y subscript 5, which you might have called y5.

The 5 informs us it's the fifth year or that's five iterations, five calculations have occurred.

Now instead of a table, we could have listed the year-on-year values using this iteration notation.

I would say y0 = 200 pounds, y1 = 212 pounds, y2, y3, y4, y5.

Why I am starting to like this iteration notation, because that's gonna be far quicker to write than having to draw that table out and write year 0, year 1.

I like efficiency.

We know that in our iterative process, our calculation, so to speak, our input from one year became the input for the next year.

The y1 value was input to find the y2 value.

The y2 value was input to find the y3 value.

The y3 value input to find the y4 value.

Quick check you've got this.

Fill in those blanks.

What's coming next? Welcome back.

Hopefully at that first blank, you had y subscript 4, y4.

The y4 value was input to find the y5 value.

And in that next sentence, if we want to know the y6 value, we would need to input the y5 value.

By repeating our calculation, we got from y1 to y2, y2 to y3, y3 to y4, y4 to y5.

The value of the digit in our subscript increases by one as we move from one iteration to the next.

We can write a generalisation from this using y subscript t to y subscript t + 1.

That's the current year's value and the next year's value.

This is our scenario again.

Our investment of 200 pounds at an interest rate of 6%.

In the original problem, we had to use repeated multiplication by 1.

06 to get from one year to the next.

We can express this relationship between the current term and the next term using an iterative formula.

There is our iterative formula.

What is it saying? It's saying the next value, y subscript t + 1 is equal to the current value multiplied by 1.

06.

Quick check you've got that.

Match the scenarios on the left to the correct iterative formula on the right-hand side.

Pause.

Give this one a go.

Welcome back.

Hopefully you matched them up like so.

Now you'll notice I left a few distractors there, a few wrong answers that looked like they might be right, but were in fact wrong.

It was very naughty of me, but it's important I put them there because it's important that you understand why they are wrong.

This scenario we'd write as y subscript t + 1 = y subscript t multiplied by 1.

04.

That's the iterative formula.

Why was it not 700 multiplied by 1.

04 = 728? Well, it's not because whilst it's the right multiplier, it's the correct first calculation, it's not an iterative formula, it's just one calculation.

An iterative formula we'll use for repeated calculations.

Our second scenario you would've matched to that iterative formula, but there were two distractors there which looked like they might be the right answer to this one, but they were not.

Why not? Well, it wasn't the first one because whilst it's the correct final valuation of the scenario, it's not an iterative formula, it's just a calculation, nothing is repeating.

The bottom one? Did you spot it? Why that one's wrong.

The current value and the next value are the wrong way around.

It's really important we use the correct notation and subscripts.

For the final one, you would've matched that scenario to that iterative formula and it wasn't y subscript t = y multiplied by 1.

025 because it's incorrect notation for current value and next value.

Pay careful attention to this.

Next, it's the same scenario again and we have our iterative formula, but something's missing.

We can add y0 when communicating our iterative formula.

What do I mean by Y0? It's our starting value.

What do we have after zero years, after zero iterations? If I gave you an iterative formula without y0, you would ask me, "Where do I start?" When I tell you y0 = 200, you know exactly where to start.

We now concisely have all the information we need to go about finding iterations in this sequence of values.

Quick check you've got that.

I'd like you to match these scenarios to the correct iterative formula.

Pause, get matching.

Welcome back.

Hopefully you matched them up like so.

Let's check why these were the wrong answers.

In the first case, it doesn't say y0.

It says y subscript t = 15,000, which means we don't know what year that 15,000 is.

That second incorrect answer should say y0 is the principle value of 3000 pounds, it's not y1.

For that last one, we didn't have a scenario with a principle value of 700 pounds and an interest rate of 5%.

You'll see a variety of letters in iteration notation.

In the context of a savings account growing year-on-year, it makes sense to use y to represent year with subscript t representing the year number, the number of iterations.

In sequences we frequently use a u to show we are referring to a term the sequence with subscript n representing the position in the sequence.

You see that in that iterative formula at the bottom there.

In the context of a population model, you might see capital P used to show we are referring to the population size with subscript t representing the year number or the number of iterations of population change.

There's an example of an iterative formula in population.

In this model, the current population decreases by 500.

That's what's happening inside that bracket.

Then the remaining population increases by 10%, hence a multiply of 1.

1, to find the next year's population.

Quick check you've got that.

True or false? Iteration notation will only ever include the letters x, y, and n.

Is that true? Is it false? Once you've decided, I'd like you to select one of the two statements at the bottom of the screen to justify your answer.

Pause and do this now.

Welcome back.

Hopefully you said false and justified that with, "We will see a variety of letters reflecting a variety of context and iteration notation, not just x, y, and n." Practise time now.

Question one, part a.

I'd like you to match these scenarios to their iterative formula.

Pause and get matching.

For part b, I'd like you to write a sentence to explain the error in the two iterative formulas you did not pair in part a.

Pause and do that now.

Question two.

I'd like you to write an iterative formula for each of these scenarios.

Pause and try them now.

Feedback time.

Question one, part a, we were matching scenarios to their iterative formula.

You should have matched these up like so.

You wanna pause and check that your matching matches mine.

Part b.

I asked you to write a sentence to explain the error in the two iterative formulas you did not pair.

For that first one that we didn't pair, you might have written, "Current value and Next value are the wrong way around." Well done if you spotted that.

For that second one, you might have said, "The formula does not specify that 500 is P zero, 500 could be any year's population." Again, well done if you spotted that.

Part two, we were writing iterative formula.

For a, you should have written y subscript t + 1 = y subscript t multiplied by 1.

06 with a y0 value of 750.

For b, you should have written that.

For c, you should have written that.

Question two, part d, getting a little bit trickier now.

The iterative formula for d should have looked like so, inside the bracket there, y subscript t - 20, that's the withdrawal of 20 pounds before the interest is applied.

You might have written it like that.

You might see it written like that.

So exact same thing, just a little more efficiently written.

Part e, this was a tricky one, would look like that.

And you might see that one written like so.

Onto the second part of our learning now.

Where we're gonna look at simplifying composite functions.

Using iteration notation makes it easier to work with composite functions which are multiple repetitions of the same function.

What do I mean by that? Well, if f(x) = 2x, a lovely simple function, we can evaluate f(5).

f(5), that's two lots of five, that's 10 We can repeatedly use this function.

We can evaluate ff(5).

Well that's ff(5) which we know to be 10.

So two lots of 10, that's 20.

We can keep this going.

I could evaluate fff(5).

That'll become f of bracket ff(5), so it's two lots of 20, it's 40.

This is another simple iterative process.

Each output becomes the next input.

The output of f(5), 10, became the input the next time and gave us an output of 20.

The output of 20 became the input, this gives us an output of 40.

So what about this one? ffffff.

I think we're gonna need to take a really close look at that one.

If we're asked to evaluate this one? ffffffffff.

How many f do you see? Can you be confident you'll carry out the right number of calculations? Iterative notation can help.

To get from f(x) to ff(x) we multiply by 2.

To get from ff(x) to fff(x) we multiply by 2.

This process will continue, it will repeat.

So we can say the next x value is the current x value multiplied by 2.

It's what that iterative formula is telling us.

The next x value is double the existing X value.

So when we come to evaluate fff, I still don't know how many f's that is.

5 is our starting value, so we can add x zero = 5, and we have an iterative formula.

We're ready to solve our problem.

Our iterative formula will write like so, x zero = 5, that's our starting value.

x1 = 10, that's f(5), it's one iteration.

X2 = 20, that's ff(5), two iterations, two repetitions of that function.

x3 = 40, that's fff(5), three iterations.

Now what about our problem, that long string of f's? Did you count 15? If so, well done.

This is going to be x15.

x15 is much easier to interpret than that long string of f's.

We want the 15th iteration.

Our calculator will give us that really quickly by just pressing the execute button 15 times if we've inputted the iterative formula correctly.

Quick check you've got that.

Which iterative formula would represent this function? f(x) = x + 7.

Pause and take your pick.

Welcome back.

Hopefully you said a.

Why was it not b? We're talking about x values as inputs and outputs from this function so we'd use an x in our iterative formula.

Why is it not c, it's incorrect subscripts.

This does not relate one x value to the next x value.

Another quick check.

By writing an iterative formula for f(x) = x + 7 you would not be asked for ffff, et cetera, you would be asked for option a, option b, option c, which one is it? Pause and take your pick.

Welcome back.

Well done.

It's option b.

x24 is much clearer than that long string of f's.

We will see more complex examples.

Let's have a look at these two.

For f(x) = 2x + 6 the iterative formula will be the next x value = two lots of the current x value + 6.

For f(x) = two lots of (x + 6) the iterative formula is the next x value is two lots of bracket, the current x value, + 6.

How about these two? f(x) = x squared + 6.

We'd use the iterative formula the next x value = the current x value squared.

For that you have to write x subscript n to the power of 2 + 6.

Then that last one, f(x) = x squared + 6x - 7.

We're gonna need to repeatedly write the current x value.

How do we do that? The iterative formula looks like this.

The next x value is the current x value squared + six lots of the current x value - 7.

Quick check you've got that.

Which iterative formula would we use for multiple repetitions of this function? f(x) = 6 lots of (x - 7.

) Pause.

Take your pick.

Welcome back.

Hopefully you said b.

x subscript n + 1 = 6 lots of (x subscript n - 7) It couldn't have been a because the current value and next value are the wrong way around, and it couldn't have been c because this formula does not reflect the priority of operations of the function.

7 is subtracted before the multiplication occurs.

Another quick check.

I'm gonna write a couple of iterative formula and ask you to repeat that skill.

I'm gonna write an iterative formula for f(x) = 5x cubed + 2.

That would be the next x value = 5 lots of the current x value cubed + 2.

Note I'm writing 5x subscript n and an exponent of 3.

For f(x) = x squared - 9x + 12, I'm gonna write that iterative formula.

The next x value = the current x value squared <v ->9 lots of the current x value + 12.

</v> Your turn.

Pause and write two iterative formula for these functions.

Welcome back.

Hopefully, for the first function you wrote that iterative formula and for the next function you wrote that iterative formula.

Pause, check that your answers match mine.

Practise time now.

For question one, I'd like you to match the functions to their iterative formula.

Pause and do this now.

Question two.

Aisha and Sam are working on this composite function.

Both pupils have made errors in what they have said.

Can you identify the errors and write a sentence explaining the error back to Aisha and to Sam.

Pause and do this now.

Feedback time.

Question one, we are matching functions to their iterative formula.

You should have matched them like so.

Just pause this video for a moment and check that your matching matches mine.

Question two, we were writing feedback for Aisha and Sam to help them improve their mathematics.

Hopefully you spotted that error in Aisha's thinking.

You might have written, "This x term should also read x subscript n to reflect the fact we're substituting the current x value into both positions in that iterative formula." For Sam, you might have spotted that is an error and you might have written "This is a miscount.

There are 12 f's, we want the 12th iteration, so this should read x subscript 12 instead." Sadly, we're at the end of the lesson now, but we have learned that using iterative formulas, we have a more efficient way of writing models of compound interest, population change, and multiple repetitions of composite functions.

Hope you've enjoyed this lesson as much as I have and I look forward to seeing you again soon for more mathematics.

Goodbye for now.