Maths C3 - Functions... Help?

Can someone please explain this example to me?
I've done all the previous sections to this from the Edexcel C3 Modular Maths textbook but this part honestly doesn't make sense to me

$m(x)=\frac{1}{x}$
$n(x)=2x+4$

$nm(x)=n[m(x)]=2[m(x)]+4=2[\frac{1}{x}]+4=\frac{2}{x}+4$

nm(x) means you take the function of m and substitute it into the function of n.

Oh right. Starting to understand a bit better now. But I think I don't really understand what the question is asking. Where is the p function answer for the first question?

Also I'm confused as surely substituting m(x)=1/x into 2/x+4 will be $\frac{2}{\frac{1}{x}}+4$ ?

It's just saying that throughout the following questions express them in terms of m(x), n(x) and p(x) where ever they may be applicable. Doesn't necessarily mean they all go into a single question.



You're not subbing it into 2/x + 4 you are subbing it into 2x+4.

2/x + 4 is not a stated function.

For part b what is an indication that you need to complete the square?

Because when you look at the functions given, the overall function cannot be m(x) because it's not a reciprocal function, and it cannot be n(x) because it's not a linear function. It only leaves you with p(x). In that function there is no $x^1$ term, thus implying the completion of the square.

Oh yeah, of course. Because 4x^2+16x+14 is a quadratic equation the 4x^2 part of the quadratic sort of hints that p(x) is one of the functions to use as it is the only function with a x^2? Is that what you're saying?

Yes. It's a quadratic and p(x) is the only quadratic function so they must go together.

Once you're comfortable with that decision, you should notice that completion of the square is the only way to get it into that form of p(x); ie (something)² - 2

This kind of question is weird and to my knowledge I've not seen it in an Edexcel C3 past paper (not to say that it can't be tested..)

The key is to understand how composite functions (i.e function of a function) work and once you're fine with what the question is asing, it's just trial and error/inspection.

Oh yeah, of course. Because 4x^2+16x+14 is a quadratic equation the 4x^2 part of the quadratic sort of hints that p(x) is one of the functions to use as it is the only function with a x^2? Is that what you're saying?

...

Ok now I'm being even more stupid. When I complete the square I get...

$4x^2+16x+14$
$=4(x^2+4x)+14$
$=4[(x+2)^2-4]+14$
$=4(x+2)^2-16+14$
$=4(x+2)^2-2$

I know $4(x+2)^2-2$ is the same as $(2x+4)^2-2$ but how come the way I've learnt to complete the square comes out as a completely different arrangement?
I'm actually having doubts as to whether to complete Maths at A2 if I can't even complete the square anymore

Then how would you do it?

A lot of students find this type of question hard. But what you've said here is the thought process you need i.e. inspection/logic/trial and error.

So e.g. for b) you notice that to get a quadratic you can put $n(x)$ into $p(x)$. And you try it and it works.

But there's no "completing the square" here. Completing the square means put $\displaystyle ax^2+bx+c$ into the form $\displaystyle a(x+h)^2 + k$.

For c), it's easy to see that $p(x)$ won't be involved so it's a case of trying different compositions of $m(x)$ and $n(x)$.

Different arrangement? How?

In the end it doesn't matter about arrangement, you still have completed the square successfully and correctly. Anything beyond that is just an extra.

How do I know what kind of functions I need to mix and match if I don't complete the square of the quadratic first?