FunctionHard
Question
It is given that f(x) is a function defined on R, satisfying f(1) = 1 and for any x ∈ R
f(x + 5) ≥ f(x) + 5
and f(x + 1) ≤ f(x) + 1
If g(x) = f(x) + 1 - x, then g(2013) equals
f(x + 5) ≥ f(x) + 5
and f(x + 1) ≤ f(x) + 1
If g(x) = f(x) + 1 - x, then g(2013) equals
Options
A.2014
B.2013
C.1
D.0
Solution
f(x) + 5 ≤ f(x + 5) ≤ f(x + 4) + 1 ≤ f(x + 3) + 2 ≤ f(x + 2) + 3 ≤ f(x + 1) + 4 ≤ f(x) + 5
⇒ In all steps there is equality only
⇒ f(x + 1) = f(x) + 1
Now f(1) = 1
⇒ f(2) = 2
f(3) = 3
f(4) = 4
:
:
f(2013) = 2013
⇒ g(2013) = 2013 + 1 - 2013 = 1
⇒ In all steps there is equality only
⇒ f(x + 1) = f(x) + 1
Now f(1) = 1
⇒ f(2) = 2
f(3) = 3
f(4) = 4
:
:
f(2013) = 2013
⇒ g(2013) = 2013 + 1 - 2013 = 1
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