What Are the Theoretical Forms for f(.) and g(.)?

[ssd]

Given that
1/ f(cx) = k - g(x) and
2/ the above is an identity,
where f(.) and g(.) are two functions
and c, k are real valued constants.

The problem is to infer upon the types of f(.) and g(.).
I have a hunch that f(.) and g(.) are logarithimic functions. Can anyone provide any analytical proof or counter example?

 

[arildno]

What do you mean by "2/above is an identity"?

 

[matt grime]

Surely there should be some mention of c on the RHS? Otherwise only constant functions work.

For f(c)=f(c.1)=k-g(1) for all c, i.e. f is identically equal to k-g(1), whatever g(1) is. And then g(x)=k-f(cx)=g(1), for all x.

 

[dodo]

Well, making c part of the definition of g(x) would do.

The obvious choice is to take g(x) = k - 1/f(cx), for whatever definition of f(x) you wish; in other words, I don't think the condition limits the actual form of the functions in any way (except by relating one to the other). For example, if f(x) = x^2, then g(x) = k - 1/(cx)^2. And so on.

Edit: Oh, sorry, I read the original condition as 1/f(cx) = k - g(x). But the argument stays the same. Just take g(x) = k - f(cx). If f(x) = x^2, then g(x) = k - (cx)^2.

(Yet another edit:)
Or... (psychic powers for the win)... by his logarithmic hunch, the OP meant something like f(cx) = h(c) - g(x), with f(x), g(x), h(x) functions and c a constant.

 

[ssd]

What do you mean by "2/above is an identity"?

We are given with two conditions. 1st condition is the expression given in point 1. The 2nd condition is given in point 2, which states that the expression given point 1 is an identity.

 

[ssd]

Surely there should be some mention of c on the RHS?

'k' in RHS can involve 'c' in some suitable form. Similarly 'c' can involve 'k' in some form. Or, g(.) may contain c or f(.) may contain k... I don't really know. I know that 'k' and 'c' are two numerical constants obtained from an experiment, and they don't change with change of the study parameters of the experiment.

 

[matt grime]

Oh, it can, can it? Are there any other things you've not mentioned?

 

[ssd]

Oh, it can, can it? Are there any other things you've not mentioned?

No, nothing is appearing in my mind at this moment except that f(.) and g(.) are real valued.

 

[dodo]

Is there a table of numeric results for f() and g(), coming out of experiments, that you can post?

 

[ssd]

Is there a table of numeric results for f() and g(), coming out of experiments, that you can post?

Yes it is there. In fact there are a number of such tables obtained from different experiment stations by different scientiests under similar set up. The values are small in number (like 10 to 12) in one set. Practically may functions fit (numerically) the data (of one set) well. There are moderate variations in data from different stations. But the problem is to find a theoritical form. One has to infer only on the basis of information I gave already. I (we) have theoritically simplified the work upto this. The usual forms of the expressions and the method (using which c & k are found in current practice) are really complicated.