# Considering a combination of functions

I am sorry for the vague title - its because I don't know what I am talking about. I have a function whose value is determined by considering the outputs of some other functions. For example,

F(x) = G(a,b) and H(x) and T(y)

• As you can see, I am not sure how to write the "and" part. That is, I want to say: In order to calculate F(x), you first have to calculate G(a,b) and H(x) and T(y); and then "mix them up in a certain way".
• How can I represent this idea mathematically? Can somone please point me to some resources that further explain this in baby-steps?

Followup:
Thank you for all your replies. Let me clarify what I want to do. Consider a scenario where I want to find the value of a car. This depends on factors such as mileage, model, number of accidents, and also owner-bias. The owner-bias itself is a function of gender, years of driving experience, and age

So.. how can I come up with a symbolic way to represent this AND also define the interralations between all these factors? I am sorry for the newbie question, please feel free to direct me to the relevant literature/terminology

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## 5 Answers

Just some possibilities to get you thinking...

If these are numerical functions, you might have:

F(x,y,a,b) = G(a,b) + H(x) + T(y)

If a and b are independent variables, you have to list them as arguments for F as above.

If, on the other hand, a=H(x) and b=T(y), you might have:

F(x,y) = G(H(x),T(y))

The possibilities are, of course, endless. I hope this helps.

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I think the OP wants a notation which is independent of the particular recipe. –  Qiaochu Yuan Jan 2 '11 at 22:14
I have updated my original post. BTW, I not only want a correct symbolic representation, but also the recipe. In other words, You are right to assume that these are all numerical functions.. I want to understand how I can come up with a recipe for these. In my original post, I said "mix them up in a certain way". How do I come up with this mixture? –  varuman Jan 3 '11 at 1:16
There is no simple mathematical formula for valuating used cars. Google "Kelley Blue Book" and get some idea of how the pros do it. –  Dan Christensen Jan 3 '11 at 5:04
My guess is Blue Book analyzes sales data to get the average selling price for each combination of factors -- make, model, year, geographic region, mileage, condition, options, etc. -- and perhaps makes short-term forecasts based on recent trends. –  Dan Christensen Jan 3 '11 at 6:13
Dan, this scenario was just an example. It has nothing to do with selling cars. I'd expect there to be a generic framework, or at least literature, for attacking such problems right? –  varuman Jan 3 '11 at 9:41

The issue appears to be one of "dummy variables" and maybe composition of functions. When you write, for example, q(x)=x^2, it really doesn't matter that the variable is x. You also have that q(y)=y^2 and the same for any variable you name. But we expect the same variables on both sides of the equation. If we could show F(x)=G(a,b) that would say that F(x) doesn't depend upon a and b, so F(x) and hence G(a,b) must be constant. If F really depends upon G(a,b), H(x), and T(y) we would expect to see F(a,b,x,y)= some expression involving G(a,b), H(x), and T(y). An example would be F(a,b,x,y)=G(a,b)*H(x)+T(y). Sometimes H(x) and T(y) are the inputs to G. That we would write F(x,y)=G(H(x),T(y))

Qiaochu Yuan and I are trying to get at the same thing using different language. I hope at least one is helpful.

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I have updated my original post with extra details. –  varuman Jan 3 '11 at 1:09

Well, if that's how you calculate $F(x)$, then it's not just a function of $x$, is it? You should write it as $F(G(a, b), H(x), T(y))$. Unless you're saying that $a, b, y$ are all themselves functions of $x$, in which case I guess you should write

$$F(x) = S(G(a, b), H(x), T(y)).$$

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I have updated my original post with extra details. –  varuman Jan 3 '11 at 1:16

To add to what has already been said, it is entirely up to you to ensure that all the "independent" variables (e.g. mileage, model, years of driving experience, age...) are indeed independent; that tends to be a sticky statistical problem. Once you are sure of which independent variables you actually need, then your function $F$ ought to have the same number of arguments.

Thus your function would be something like

$F(\text{mileage},\text{number of accidents},B(\text{gender},\text{age},\text{driving experience}))$

where $B$ is the function for owner bias... you get the idea.

Another thing: figuring out how to combine these independent variables all together is your burden, in that you have to experiment on/derive how to properly relate all your variables.

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(Note to webmaster: The Add Comment function doesn't seem to be working, so I am submitting another answer.)

Varuman, if you are going to look at several difficult to quantify variables, the statistical approach (my Blue Book example) is widely applicable. If you are determined to get a formula, look up regression analysis. It presents a number of techniques for entering data points and coming up a formula of best fit. Of course, you have to have some idea of what the formula will look like and what the variables are. You can assume, for example, that the function is a linear combination of the independent variables, e.g. y = a0+ a1*x1 + a2*x2 +... where y and the xi are variables for which you enter values and the ai are parameters that the software will calculate.

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