Let
f
:
R
→
R
be a function that satisfies the following property.
For all x ∈ R , f ( x ) 2 = x 2 .
Consider the following statements.
[ 1 ] . f ( x ) has to be equal to x for all x ∈ R .
[ 2 ] . f ( x ) has to be equal to − x for all x ∈ R .
[ 3 ] . In fact, f ( x ) could be one of three things. f ( x ) = x for all real x , f ( x ) = − x for all real x and f ( x ) = ∣ x ∣ for all real x
[ 4 ] . It is impossible to tell what f ( x ) is.
Which of these statements is correct?
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Nice question. I misread the third option, overlooking the "for all x" part. Oh well...
Just another comment. If continuity is mentioned, then the third option is the correct one, I think.
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Nope, even then, the third option is incorrect. Remember ( − ∣ x ∣ ) 2 = x 2 ?
Actually it is possible to tell what f is, in the sense that we give a complete characterization of all possible f .
Partition the reals into two subsets A , B arbitrarily. Let f ( x ) = x ∀ x ∈ A and f ( x ) = − x ∀ x ∈ B . This is all the solutions.
To see this, observe that f ( x ) 2 = x 2 implies ∣ f ( x ) ∣ = ∣ x ∣ , so f ( x ) = x ∨ − x . However, there is no way to relate f taken on two different values from just the given equation, so the best we can do is that, f ( x ) = x ∨ − x for all x . The above simply formalizes this, choosing which give positive values and which give negative values.
Since for each real number besides 0 we have two choices for the function, in total we have 2 c such functions where c is the cardinality of the real numbers. Georg Cantor proved that ω < 2 ω for any cardinal number ω , and in particular c < 2 c ; thus we have more functions satisfying this condition than there are real numbers!
This is an awesome solution specially because of the last paragraph. Voted up!
Consider the function f ( x ) = x when x is rational, and f ( x ) = − x when x is irrational. This function satisfies the equation but does not satisfy [ 1 ] , [ 2 ] , or [ 3 ] .
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It is impossible to say what f ( x ) is.
f ( x ) could be equal to x in certain intervals while it could be equal to − x in other intervals.
Consider this example.
f ( x ) = { x , if π ≤ x ≤ 2 π − x , if x < π or x > 2 π . .
In fact there are infinite possibilities for f ( x ) and it is impossible to tell what f ( x ) is without any additional information.