Are All Functions Dependent on Each Other?

[Heirot]

When are to functions y₁ = f₁(x) and y₂ = f₂(x) independent? It would apper never, because, we can always write x = f₁^-1 (y₁), and therefore y2 is a function of y1. Every function is dependent of any other function. Generally, dy₁/dy₂ != 0 for arbitrary functions y₁ and y₂. Is this reasoning correct?

 

[Ben Niehoff]

We can, however, talk about linearly-independent functions: namely, functions with vanishing Wronskian.

 

[Defennder]

I can't figure out what the OP is saying about independent functions. It's not the same as the concept of linear independence, is it?

 

[Mute]

We can, however, talk about linearly-independent functions: namely, functions with vanishing Wronskian.

It's when the Wronskian doesn't vanish that the two functions are linearly independent.

 

[HallsofIvy]

When are to functions y₁ = f₁(x) and y₂ = f₂(x) independent? It would apper never, because, we can always write x = f₁^-1 (y₁), and therefore y2 is a function of y1. Every function is dependent of any other function. Generally, dy₁/dy₂ != 0 for arbitrary functions y₁ and y₂. Is this reasoning correct?

I have no idea what you are saying! Do y₁, y₂, x₁, x₂ just represent numbers? What, then, is the difference between saying y₁= f1(x₁) and just y= f1(x)? And, of course, what do you mean by "independent"? Apparently you don't mean "linear independence". Before anyone can tell you whether or not "any two functions are not independent" you will have to say what you mean by two functions being "independent"!