Stanton D. answered • 03/28/14

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Crystal, what he's saying is, you've quoted a question (problem) without giving any indication of what the question refers to. It's like asking us, "What's in this picture?" but not showing the picture!

So, when the question says "operations that are performed on the functions", you have to show us the examples that were given with the question originally. Or else, as Steve suggests, look at the bunch of examples you were originally given with the question, and look for evidence that the expressions in the examples used addition, subtraction, multiplication, or division ANYWHERE in them. If they did, then you can re-write them in the forms that Steve quoted. For example, if you were given the expression (2x^2 - 4x +2) you can re-write that as (2x^2) - (4x - 2) and say that that is f(x) - g(x), where f(x) = 2x^2 and g(x) = 4x - 2 . You have to be able to shift back and forth between explicit expressions (2x^2) and representing those same things as function expressions (f(x), in this example). How are these ways of writing the same thing different? Well, when you say "f(x)", then that could stand for 2x^2 (as it does here), but it might also stand for other things, when you need it to.

Maybe your issue is, you don't understand why you might want to go from something you pretty much recognize (2x^2) to something that seems vague (f(x))?

That's because some of the things you do in math to solve equations, fall into patterns, that you can learn to recognize as patterns, and solve the same general way for the same pattern.

Here's an example: "Clear the square root from the denominator of the expression:

(2 + 3x)/(x - 2(x)^0.5)"

Now, are you likely to have ever seen this *exact* problem before, so that you remember the *exact* thing you did before? Not likely!

So, instead, you look for a pattern, and you find one: the denominator has the form (a-b), where a=x, and b=2(x)^0.5 .(For this problem, these are like f(x) and g(x) respectively, aren't they). And by now you should know, that one of the things you can do with the expression (a-b), is multiply it by (a+b), to get (a^2 - b^2). This will give you a result that gets rid of the (x)^0.5 part of the denominator. So:

=(2+3x)(x+2(x)^0.5)/(x-2(x)^0.5)(x+2(x)^0.5) = (2x +3(x)^2+4(x)^0.5+6(x)^1.5)/(x^2-4x) is your result.

By the way, you also do something a little like this, whenever you set up an equation (2x=4) to solve a problem, ("Mary runs 4 miles in 2 hours. At the same speed, how far does she run in one hour?") . Here, x stands for (it's an abstraction for) some value you need to find out (but don't know yet) in your problem; all you know is that two of them must add up to the value 4! You then can go on to solve the equation, obtaining x = 2 miles (I hope!). What's the value of representing your original problem as 2x = 4? First of all, it shows that you transformed the word problem correctly into a mathematical equation. You need to be able to do this of course in able to solve any word problem! But also, it shows you the type of equation that could be used to solve ANY word problem of the same sort as your particular one, that might have different particular values quoted in it.

In the same way, for lots of the things you will need to do with more complicated expressions eventually, you will need to recognize patterns in order to apply the rules you will learn. We just call the items in the patterns things like f(x) and g(x), in order to keep track of them. It's a little like looking for your other shoe in the morning, you say, "Have you seen my right shoe?", not "Have you seen Marjorie?" (if you name each of your shoes individually, and Marjorie happens to be the name you gave that particular shoe). Everyone can then start looking for your right shoe (f(x)); you don't have to explain what Marjorie looks like: red, with a bow, and (2x^2 - 3x -4) written on it, fits the right foot.

Hope this puts it in a bit of perspective for you!

Crystal J.

03/27/14