WEBVTT

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Hi I'm Fiona.

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Let's have a look at Timura 2021 paper
one and question 5.

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We're given a function F of MN which has
two function rules.

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F of m * F of N is the output if m * n is
a multiple of 3 and m * n which is the

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input,
is the output as well if m * n is not a

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multiple of three.

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We're told M&
N are both positive integers and we are

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given F of 9 + F of 16 -,
F of 24 = 0 and asked what is the value

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of F of three?

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We've got our options here,
A to F to choose from once we get to an

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answer.

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So this question here involves A
piecewise function that has two different

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function rules depending on the input and
whether it's a multiple of three or not a

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multiple of 3.

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Now we need to find the value of F of
three.

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So let's just use the function rule 1st
to see what happens with that.

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So 3 is a multiple of 3.

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So I need to think about this function
rule.

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The only way I can split three up because
it's prime is by thinking of it as 3 * 1.

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But that would give me F of 3 * F of 1
and then one is not a multiple of 3.

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So that would give an output of 1 and I
would be back where I started at F of

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three.

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So that gets me nowhere.

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And that tells me that I'm going to need
to use this result here in order to find

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a value of F of three.

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So let's do that.

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F of nine, Well, nine is a multiple of 3,
and we can think of it best as a power of

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three.

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It's 3 ^2.

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If I just think of it as 9 * 1, again,
I will not get anywhere because I've

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tried that already.

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And so I'm going to think of F of nine as
F of 3 * 3.

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And then my output becomes F of 3 * F of
three.

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Now for the next term,
in this result that we're given,

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16 is not a multiple of 3,
and so F of 16 will just give an output

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of 16 itself.

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Now I have minus F of 24.

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I'll deal with the -, in a bit,
but now just thinking of F of 24, well,

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24 is a multiple of 3,
so I'm going to need to use this part of

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my function rule.

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I'm thinking about 24 as a factor of 2
numbers.

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The most helpful thing to do here is to
take out any powers of 3/24 is 3 * 8.

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So it just has one power of three in it.

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So I'm going to think of 24 or F of 24 as
F of 3 * F of eight.

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Now, F of three,
we're not going to do anything with that

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as yet, but F of eight,
8 is not a multiple of 3,

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so it's output will just be 8 itself.

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So this is the same as 8 * F of three.

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And coming back here,
I have that minus F of 24 is the same as

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-8 F of three, and that all equals 0.

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And now I'm starting to see that there is
an underlying structure here that is

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quite typical of Tamima.

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So if you've come across this before,
you will have in your mind what I've

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spotted too,
which is that this is a quadratic in F of

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three.

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So I have F of 3 ^2 -8 F of 3 + 16 = 0.

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And this is going to be quite
straightforward to factorise and then to

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find values for F of three.

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So factorising this quadratic a +16 is
the same value as -4 * - 4 and -4 -,

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4 adds 2 - 8.

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So that's going to be my factorization
and this will give me a repeated solution.

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And in order for this to be true,
then it has to hold that F of 3 = 4.

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Now looking at the options that I have to
choose from,

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I can see that option F gives a value of
four.

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So that's my answer for this question.