# Is it theoretically possible to find g(x) from this equation?

• Zomboy
In summary, the conversation discusses the possibility of finding the function g(x) from the given equation. The notation is critiqued and it is concluded that g(x) does not exist in this case. It is also discussed whether a function of (x,y,z) and a function of (x) can give a function of (y,z), with the conclusion being that it is possible. However, the original problem has no sensible solution.
Zomboy
Is it theoretically possible to find "g(x)" from this equation?

So through my workings on this question I came up with this equation:

(x^2)yz - (y^2)(x^2) - x + g(x) = g(y,z)

* where g(x) is some function of x and g(y,z) is some function of y and z

I'd like to derive g(x) from this although I get the feeling that it simply can't be done. Can say a function of (x,y,z) + a function of (x) ever give you a function of (y,z)? Why can't you do this?

Your notation is bad. When you write g(x), it implies that is a function in one variable. But g(y,z) is a function of two variables. This is impossible.

Are you saying the g(x) and g(y,z) are different functions (in which case, you shouldn't use the same letter!) or that g(y,z) is "the same function as g(x) except using two variables" (which is meaningless).

Furthermore, (x^2)yz - (y^2)(x^2) - x is a function of three variables, x, y, and z.

Notation is bad, but we should be able to interpret g(x) and g(y,z) as two different functions, the point being that the first depends only on x, while the latter depends on y and z.

Such a g(x) does btw not exist in your particular case. For example, y=z=0 gives g(x)=x+C, and y=1,z=0, gives g(x)=x2+x+D, where C and D are some constants.
Can say a function of (x,y,z) + a function of (x) ever give you a function of (y,z)?
Yes, trivially, for example (x+y+z) + (-x) = (y+z).

Well, if you meant that g(x) and g(y,z) are different functions, then you would get:
(x^2)yz - (y^2)(x^2) - x + g(x) = h(y,z)
Its partial derivative with respect to x is:
2xyz-2xy^2-1+g'(x)=0

Since this is an expression with y and z in it, this means that there is no such g'(x) that depends only on x.

So we have to assume that you actually meant g(x,y,z) with depends on all of them.
In that case, you get:
$$2xyz-2xy^2-1+{\partial g \over \partial x}(x,y,z)={\partial g \over \partial x}(x,y,z)$$

But then, this would only be true if ##2xyz-2xy^2-1=0##.

So the question becomes: what did you actually mean?
The problem that you state has no sensible solution however it was meant.

## 1. Can g(x) be found from any equation?

No, not all equations allow for the solution of g(x). It depends on the complexity and structure of the equation and whether it has a unique solution for g(x).

## 2. What factors determine if g(x) can be found from an equation?

The factors that determine if g(x) can be found from an equation include the number of variables, the degree of the equation, and the availability of constraints or boundary conditions. These factors affect the solvability of an equation.

## 3. Are there certain types of equations that make it easier to find g(x)?

Yes, there are certain types of equations that are more easily solvable and therefore make it easier to find g(x). These include linear equations, polynomial equations, and certain types of differential equations.

## 4. Is it possible to find g(x) if the equation is too complex?

Yes, it is still possible to find g(x) from a complex equation, but it may require more advanced mathematical techniques and tools. In some cases, it may not be possible to find an exact solution and approximations may be used.

## 5. Can g(x) be found if the equation has multiple solutions?

If an equation has multiple solutions, it may be possible to find g(x) depending on the specific solution being sought. However, if the equation has an infinite number of solutions, it may not be possible to find g(x) without additional constraints or information.

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