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/
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