I'm trying to find the function with some properties:
1. f(u) = 0 iff v = 0
2. f(av) = |a| f(v)
Does these kind of functions exist and if yes, how do they called and where can I get more info about it?
Thanks in advance!
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I'm trying to find the function with some properties:
1. f(u) = 0 iff v = 0
2. f(av) = |a| f(v)
Does these kind of functions exist and if yes, how do they called and where can I get more info about it?
Thanks in advance!
Название |
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Yes. f(x) = |x|
And i think that first property is redundantly since f(0) = f(0 * x) = |0| * f(x) = 0yeah
f(x) = 0
The "iff" in the first property means "if and only if". This is an equivalence property which can be rewritten as
If u = 0, then f(u) = 0.
If f(u) = 0, then u = 0.
The contrapositives of these two conditional statements are:
If f(u) ≠ 0, then u ≠ 0.
If u ≠ 0, then f(u) ≠ 0.
Here is short solution to your functional equation:
Let f(1) = c for some constant c ≠ 0. Then using (2), f(a·1) = |a|·f(1) = c|a|. Therefore, functions which satisfy the properties must be of the form f(x) = c|x| .
Checking these solutions, it is clear that f(0) = c|0| = 0 and since c ≠ 0, (1) is satisfied. Additionally, f(av) = c|av| = c|a||v| = |a|f(v) by property of absolute values.
Therefore, all
work.
Notably, f(x) = 0 (as proposed by farmersrice) does not work as it fails the "only if" portion of (1).
Additionally, the first condition is not redundant for the same reason: without it, we may only conclude the "if" and not the "only if" from (2).
shoot, you're right!!!! I missed the last f