Discussion Problem_(25)

[duck]

Q1) Let, f:(-1,1) → (-1,1) be continuous such that f(x) = f(x^2) for every x and f(0)=1/2. Find the value of f(1/4).

[Sinistral]

1/2

[MI]

Hi ,

I am getting 1/2. But its more on intuition.

As f(x) = f(x^2) = f( x^4) =   so on....

and as |x|< 1   .....f(x^very high number) approaches to zero and continuity condition gives the answer.

Duck and Sinistral, is there any mathematical proof?

[diptendra1991]

1/2 because the graph is actually a straight line parallel to x axis and that crosses the y axis at 1/2.

[Sumit]

Ans.1/2.

[duck]

Correct Answer = 1/2

@MI:
We're just given f(x)=f(x^2) for every x.
Nothing is said about f(x^4) and so on.. So, dont take that as given.

[Ayushya Kaul]

Hi Duck,
Could you please explain the answer?