What exactly do you mean by f~g in this context? ( what kind of equivalence)
Hello!
I hope this is the correct place to ask this.
So, I need to find out this :
If f~g then f'~g'? Where x seeks to 0, and f,g - functions, that can be diferentiated.
Checked some examples with functions and they all gives positive answer to the question. But cannot find how to proove it in general, or maybe it is impossible?
well, i hadn't thought about it. but in way i look at this task, f~g if dom(f)=dom(g), ran(f)=ran(g) and f(x)=g(x).
example: f(x)=sin(arccos(x))
g(x)=sqrt(1-x^2), both of them in my opinion are equivalent.
f'(x)=(sin(arccos(x)))'=-x/sqrt(1-x^2)
g'(x)=(sqrt(1-x^2))'=-x/sqrt(1-x^2), those obviously are equivalent.
which is the another way about equivalent functions and what do you say about this?
Well yes, if that's what you mean, those two functions are identical in the litteral sense. What you said is in fact nothing more then stating:
Ofcourse, you can define a equivalence relation on a set of functions: Something like f~g iff f(x)=g(x) almost everywhere (that is, these functions may disagree on a finite (or countable set of points).which is the another way about equivalent functions and what do you say about this?
Except that, as functions, x is NOT equal to .
The function, x, is defined for all x, the function, is defined for all x except 0. They are not the same function because the have different domains.
Ofcourse, you can define a equivalence relation on a set of functions: Something like f~g iff f(x)=g(x) almost everywhere (that is, these functions may disagree on a finite (or countable set of points).
You have still not made clear what you mean by "~". You said before "well, i hadn't thought about it. but in way i look at this task, f~g if dom(f)=dom(g), ran(f)=ran(g) and f(x)=g(x).
example: f(x)=sin(arccos(x))
g(x)=sqrt(1-x^2), both of them in my opinion are equivalent."
In that case, f and g are the same function, regardless of what "formula" you use to represent them.
sin(arccos(x)) and are just different ways of writing the same function.
What is true is that if f(x)= g(x) on some interval around , then because the derivative is a local property.