341If a number a is in the range of f, then f(a) = 1/a. Does 500 belong to the range of f?
if a
continous
function satisfies F(F(x))=1/F(x) and F(1000)=999,
Find F(500)
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12 Answers
341
1yes, it was given that the function is continous from R→ R
okay Let me give some options:
A) 500
B)1/500
C)499
D)1/499
E)501
F)1/501
66If I remember correctly, this was asked in Leningrad Mathematical Olympiad 1988. An interesting one requiring the use of the Intermediate value property of continuous functions.
341@kaymant: that's right. and i remember someone on this forum answered this on the same lines as the hint i gave. I am not able to trace that post
71Ohh. Yes .. Got it.. I misinterpreted some numbers.
Thanks Sir and SUbho!
1Replace x with x+1 ...
f(x)+f(x+2)=3.5f(x+1)
then with x-1,
f(x)+f(x-2)=3.5f(x-1)
add both
f(x+2)+f(x-2)=f(x)
now replace x with x+2, then with x-2,then add both
we get
f(x)+f(x+6)=f(x+2)+f(x-4)
now proceed