composite function in graphs? (1 Viewer)

[sam_i_am_1095]

does anybody know if this is actually part of the syllabus? its in the cambridge text, but our teacher just skipped over it. should i look over it, or is it a waste of time?

 

[Carrotsticks]

does anybody know if this is actually part of the syllabus? its in the cambridge text, but our teacher just skipped over it. should i look over it, or is it a waste of time?

Most certainly part of the syllabus.

You have problems in the form f(g(x)) in the exam, as well as g(f(x)).

For example:

"Here's f(x), draw 1/f(x)", which is actually g(f(x)), where g(x) = 1/x

You also have this:

"Here's f(x), draw f(x^2)", which is actually f(g(x)), where g(x) = x^2.

However, the latter has only been asked once in the HSC recently (2009 HSC).

 

[braintic]

Most certainly part of the syllabus.

Actually, in its general form, graphs of composite functions is not mentioned anywhere in the syllabus.

The syllabus mentions only a few specific types: y=1/f(x), y=[f(x)]^n, y=sqrt [f(x)]. And it does not say that these are merely examples of what can be asked - they each have their own heading, and there is no elaboration beyond these specific types.

But Extension 2 exams are notorious for going beyond the scope of the syllabus, and the more general questions have been asked on numerous occasions.

 

[Trebla]

Actually, in its general form, graphs of composite functions is not mentioned anywhere in the syllabus.

The syllabus mentions only a few specific types: y=1/f(x), y=[f(x)]^n, y=sqrt [f(x)]. And it does not say that these are merely examples of what can be asked - they each have their own heading, and there is no elaboration beyond these specific types.

But Extension 2 exams are notorious for going beyond the scope of the syllabus, and the more general questions have been asked on numerous occasions.

It is not explicitly mentioned but graphing composite functions fall under:

1.8 General approach to curve sketching
The student is able to:
• use implicit differentiation to compute dy/dx for curves given in implicit form
• use the most appropriate method to graph a given function or curve.

In other words, if I give you any random function you should be able to sketch it. Of course, under exam conditions with limited marks, composite functions is probably the easiest way to assess this without consuming too many marks.

 

[braintic]

It is not explicitly mentioned but graphing composite functions fall under:

1.8 General approach to curve sketching
The student is able to:
• use implicit differentiation to compute dy/dx for curves given in implicit form
• use the most appropriate method to graph a given function or curve.

In other words, if I give you any random function you should be able to sketch it. Of course, under exam conditions with limited marks, composite functions is probably the easiest way to assess this without consuming too many marks.

By that logic, absolutely anything could be asked in the HSC.

The only reason we know that composite functions can be asked is that it has been asked before, not because of any syllabus reference.

 

[Carrotsticks]

By that logic, absolutely anything could be asked in the HSC.

Yes, that is correct.

The HSC is shifting away from strict syllabus definitions (and it has done so for a while now, case in point being conics locus problems) and recent exams have contained 'non-syllabus' topics.

For example, f(1/x), is that in the syllabus? (2009 HSC Q3)

A telescopic sum (2010 HSC Q8)

The squeeze law (2010 HSC Q8)

Second order homogenous differential equations (2011 HSC Q4)

The floor and ceiling function (2012 HSC Q16)

There are various other questions that do not exclusively have names, but they still stray from the syllabus.

MX2 2011 Q4 (a) Locus Problem: Find locus of |z-a|^2 - |z-b|^2 = 1

However, if you look at the syllabus, I quote from it:

The student is able to:
• given equations Re(z) = c, Im(z) = k (c, k real), sketch lines parallel to the
appropriate axis
• given an equation | z – z1 | = | z – z2 |, sketch the corresponding line
• given equations | z | = R, | z – z1 | = R, sketch the corresponding circles
• given equations arg z = q, arg(z – z1) = q, sketch the corresponding rays
• sketch regions associated with any of the above curves (eg the region
corresponding to those z satisfying the inequality (| z – z1 | ² R)
• give a geometrical description of any such curves or regions

I do not see anything in the form of the 2011 locus problem provided in the syllabus.

Furthermore, check out the 2011 HSC conics problem Q5 (c) (iii), which is "Hence or otherwise, prove that Q lies on the circle ...."

This looks an awful lot like a locus problem to me, yet in the HSC they specifically state

Locus problems on the ellipse are not included.

What I'm saying overall is that if we study the MX2 course strictly bounded by the syllabus, we will not get very far. The idea is to teach students how to think, not how to do. The current syllabus is teaching students to 'do' (hence why they state specific types of questions) and it's slowly shifting away from that mentality, to something more true to how I think (and hopefully you too) Mathematics should be taught.

So to summarise, I think students should always be prepared to have things thrown at them that are not strictly in the syllabus, but have some sort of INTUITIVE notion behind them. This relates back to what I was saying before about teaching students to THINK, not DO.

The only reason we know that composite functions can be asked is that it has been asked before, not because of any syllabus reference.

Where has it been asked before? Curves in the form , where f(x) is provided, have been asked plentifully, but I'm interested to know when curves in the form have appeared previously.

Also, I'm not following your logic here.

The only reason we know that X can be asked is because it has been asked before

X is some topic that is new. Going backwards with your logic (kinda like induction), there must have been some point where X was asked without having been seen in past HSC exams (kinda like n=1).
What do we do then to prepare for that?

 

[asianese]

What I'm saying overall is that if we study the MX2 course strictly bounded by the syllabus, we will not get very far. The idea is to teach students how to think, not how to do. The current syllabus is teaching students to 'do' (hence why they state specific types of questions) and it's slowly shifting away from that mentality, to something more true to how I think (and hopefully you too) Mathematics should be taught.

^!!!

 

[Trebla]

By that logic, absolutely anything could be asked in the HSC.

That's right, anything can be asked in the HSC provided you have the conceptual tools within the syllabus content to answer them (i.e. you don't need to know anything outside the syllabus to solve it).

A simple example of this, partial fractions with repeated factors technically is not in the syllabus so the following question is outside the syllabus:

Split the following into partial fractions



HOWEVER, partial fractions with repeated factors do appear in the HSC exams. Why? Because the questions are often worded as below which make it within the scope of the syllabus:

Find the values a, b and c such that



Note that in this case, the form of the partial fraction split is GIVEN whereas in the earlier case it was not given. In other words, you are not expected to know what the partial fractions for an expression containing a repeated factor should look like (which is a conceptual thing that is not in the syllabus that is needed to solve the earlier question), BUT you should be able solve the problem with the form given to you using standard algebraic techniques which is well within the scope of the syllabus (since you're not required to recall the form of the partial fraction split in this question).

 

[braintic]

I agree with everything you just said.

BUT the extended composite functions questions are not covered by the logic of that example. There has been no lead-in question in past HSC questions [ for example f(x^2) ]. There is no part of the syllabus which prepares students for this example.

 

[Trebla]

I agree with everything you just said.

BUT the extended composite functions questions are not covered by the logic of that example. There has been no lead-in question in past HSC questions [ for example f(x^2) ]. There is no part of the syllabus which prepares students for this example.

There is nothing outside the syllabus that you need to know conceptually to graph f(x²).

Even when you graph say 1/f(x), you're simply investigating the effect of taking the reciprocal of the ordinates on the features of f(x) (e.g. formation of asymptotes, effects on turning points etc). It's basically applying everything you already know about curve sketching from Extension 1 so there really is nothing new conceptually speaking as you already have the tools to be able to sketch it. It just so happens to be a mandatory example in the syllabus.

Same goes for composite functions. For f(x²) you can investigate how the turning points behave, symmetry of the graph (noting it is an even function) and how the graph behaves at extreme points. Nothing in this investigation is conceptually new. You're just applying something you already know in an unfamiliar situation.

 

[Carrotsticks]

There is nothing outside the syllabus that you need to know conceptually to graph f(x²).

Even when you graph say 1/f(x), you're simply investigating the effect of taking the reciprocal of the ordinates on the features of f(x) (e.g. formation of asymptotes, effects on turning points etc). It's basically applying everything you already know about curve sketching from Extension 1 so there really is nothing new conceptually speaking as you already have the tools to be able to sketch it.

To add on to this, a perceptive student will notice that they can convert a f(1/x) problem into a 1/f(x) problem simply by rotating the page 90 degrees clockwise (or anticlockwise), labelling the new curve 'F(x)', then proceeding with the problem as usual so f(1/x) becomes a normal 1/f(x) problem, f(x^2) becomes a normal sqrt[f(x)] problem etc.

 

[braintic]

f(1/x) becomes a normal 1/f(x) problem, f(x^2) becomes a normal [f(x)]^2 problem etc.

Agree with the first statement, completely disagree with the second.
y=f(x^2) becomes a normal Sqrt [f(x)] problem.
And even that only works for positives.

 

[Carrotsticks]

Agree with the first statement, completely disagree with the second.
y=f(x^2) becomes a normal Sqrt [f(x)] problem.
And even that only works for positives.

Yep, my fault. Totally forgot about the fact that swapping x-y axes requires the inverse, so f(e^x) curves 'become' ln(f(x)) curves etc.