Composite Functions

Date: 01/11/98 at 10:05:00
From: Nutty
Subject: Composite functions

Dear Sir,

I'm a Math teacher and I have a problem with my students. They can't 
understand composite functions. Can you explain?

Yours sincerely,
Nutty

Date: 01/12/98 at 13:11:40
From: Doctor Joe
Subject: Re: Composite functions

Dear colleague,

I am a high school lecturer myself and I have encountered many 
problems teaching composite functions. My best friend, a fellow 
teacher has just shared with me some methods of teaching them, so I 
don't deserve all the credit for this answer. I hope the following can 
help you.

1. Induction activities

My friend and I agree that composition of functions is best understood 
by relating the concept to daily life. For instance, we find it quite 
natural to relate composite functions to a sequence of processes. 
A raw product is the input, and we first need to change it or refine 
it to obtain a secondary product. Then this secondary product is 
further refined and processed to finally obtain the desired product.  

Now call the first procedure F and the second refining procedure G.  
Suppose the raw material is x. Then we can draw the following diagram:

   x  ----->   Fx   ----->   G(Fx)

Another example: you can ask students to play with plasticine or clay. 
Each student modifies the other student's artwork; you should mention 
that the sum (composite) of their actions (one after another) brings 
about the shaping of the final artifact.

Another example that we brainstormed was the food chain. Recall that 
all of us have studied the animal-plant food chain somewhere in  
elementary school. Take grass (g) as the first input; then the cow 
(c), being a function, "eats" the grass. Next, here comes a third 
animal, say the tiger (t), and gobbles the cow. The best way to denote 
this is:
                   t(c(g))

the brackets denote the walls of the stomachs of those animals.  (It 
sounds a bit crude, but it really makes sense!)

2. Notation

Cambridge (and in fact a lot of the world) still sticks to the 
conventional notation of composition. Let the students know that 
there are some other notations, so that they need to watch carefully 
while reading other books. What we have conventionally is called the 
left notation of writing function. There is yet another to write f(x): 
namely, (x)f. In this case, f acts on the right of x and is called the 
right notation. In both cases, the rationale is the same: f is close 
to x and so acts on it directly. So, in both notations, fg(x) or (x)gf 
still displays the fact that g acted on x first and then the result is 
fed into f. 

I feel that the animal-plant food chain is quite helpful in 
remembering the notation of composition.

3. Checking the definition of composition

It's good sometimes to try this:

Let f be the procedure of planting an apple seed to yield another 
apple seed. Let g be the same, except for an orange seed. Now try to 
figure out a meaning of fof(an apple seed) (f^2 in some books). 

Well, one possible meaning is to say: Plant an apple seed to yield an 
apple seed and again plant this new apple seed to get a third one.  
Makes sense!  But this won't go well if we just try to make up a 
composition of f and g i.e. fog (for obvious logical reasons that you 
should see by now!).

Golden rules:

A) Try not to throw in all the formulae for the functions; i.e. the 
mapping rules. Just use daily examples to illustrate.

B) Bear in mind that fog makes sense if range(g) is a subset of 
domain(f).

Happy teaching.

-Doctor Joe,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/