perplexus.info :: Just Math : Function Existence Exercise

Function Existence Exercise (Posted on 2015-10-11)

R denotes the set of real numbers. Does there exist functions F: R → R such that:
F(F(x)) = x² - 2?
If so, find all such functions.
If not, prove that no such function can exist.

Source: American Mathematical Monthly

No Solution Yet Submitted by K Sengupta

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re(2): Almost there?

| Comment 3 of 4 |

(In reply to re: Almost there? by Steve Herman)

Sorry if I skipped some steps.

1) Suppose f(2)=a, then f(f(2))=f(a)=2

Since f(f(a))=a^2-2

we have a^2-2=a

solutions a=2 or a=-1

So either f(2)=2 or f(2)=-1

2) f(-2)=2 because f(f(f(-2)))=f(f(x)) which implies x^2-2=2

so x=2 (case 1)

or x=-2 (case 2) ***I may have made an error here***

Posted by Jer on 2015-10-14 10:07:48

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