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I am new to functions and domains and I am not sure that I can assume following because I think that the range of first function is $[0, 1]$ and the range of second is $(-\infty, \infty)$.

The original question: Find $f(x)$ if $f( \cos^2(x) ) = \cos^2(x)$

Asaf Karagila
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user2648841
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    That is a totally bizarre question. Are you sure you haven't omitted anything? – MJD Aug 03 '14 at 22:21
  • Check composition of functions. – Mhenni Benghorbal Aug 03 '14 at 22:30
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    Since the range of $\cos^2x $ is $[0,1]$, for any $ t $ in this range we indeed have that $ f (t)=t $. However, for values of $ t $ less than $0$ or bigger than $1$, we have no information about $ f $. That is to say that any $ f $ whose restriction to $[0,1] $ is the identity satisfies the given equation. – Andrés E. Caicedo Aug 03 '14 at 23:43

2 Answers2

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No. All you can deduce is that $f(x) = x$ for $0 \le x \le 1$, since that is the range of values that $\cos^2(x)$ takes on.

marty cohen
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You cannot assume that, unless you know more about the function. For example, perhaps $f(x)=\operatorname{min}(x, 1)$.

Micah
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