Concerning the formal definition of Big Oh.

f(n) = O(g(n))

Why is it not:

f(n) = O(f(n))

or

f(n) = O(c*f(n))

since for the Big Oh analysis, f(n)=2n and g(n)=n are identical?

I am confused on defining the function f(n) using another function

UPDATE (Based on answers):

Why isn' t the definition as follows:

f(n) <= c*abs(g(n))

What does the formal O(g(x)) offer more to be used in the definition?
From my point of view it complicates it without offering any intuition

Consider the formal definition:

f(n) = O(g(n))

Why is it not:

f(n) = O(f(n))

or

f(n) = O(c*f(n))

since for the Big O analysis, f(n)=2n and g(n)=n are identical?

I am confused by the function f(n) using another function.

Update

Why isn' t the definition as follows:

f(n) <= c*abs(g(n))

What does the formal O(g(x)) add to the definition? It seems like it overcomplicates things.

Concerning the formal definition of Big Oh.

f(n) = O(g(n))

Why is it not:

f(n) = O(f(n))

or

f(n) = O(c*f(n))

since for the Big Oh analysis, f(n)=2n and g(n)=n are identical?

I am confused on defining the function f(n) using another function

Concerning the formal definition of Big Oh.

f(n) = O(g(n))

Why is it not:

f(n) = O(f(n))

or

f(n) = O(c*f(n))

since for the Big Oh analysis, f(n)=2n and g(n)=n are identical?

I am confused on defining the function f(n) using another function

UPDATE (Based on answers):

Why isn' t the definition as follows:

f(n) <= c*abs(g(n))

What does the formal O(g(x)) offer more to be used in the definition?
From my point of view it complicates it without offering any intuition

Concerning the formal definition of Big Oh.

f(n) = O(g(n))

Why is it not:

f(n) = O(f(n))

or

f(n) = O(n*f(n))

since for the Big Oh analysis, f(n)=2n and g(n)=n are identical?

I am confused on defining the function f(n) using another function

Concerning the formal definition of Big Oh.

f(n) = O(g(n))

Why is it not:

f(n) = O(f(n))

or

f(n) = O(c*f(n))

since for the Big Oh analysis, f(n)=2n and g(n)=n are identical?

I am confused on defining the function f(n) using another function