@MI: I don't think f(x) should be constant function only it can be f(x)=x^2+1 like taani mention...this function has domain=real no. and range=rational no..however answer will be same in both the case i.e even function...plz let me know If i'm wrong...
Delhi School of Economics
"one" of the ways in which we can construct such a function is:
f(x)= 3x+1, if x belongs to Q
= 0 , elsewhere.
now, it wont be even or odd.
it definitely wont be continuous because we can always find an irrational no between any two rational numbers.
now we can construct specific examples wherein the function CAN become even (I cant right now think of a way to make such a function odd). Nevertheless, in general we cant comment whether it will ALWAYS be even or odd.
in this case it is increasing but then we can always make it decreasing (eg f(x)= -3x +19, x belongs to Q; = 0 e.w.).
so the only certain thing about a function whose co-domain is Q is, it is never continous. but its not an option.
the only feasible correct option then (according to me) should be (D) nothing can be said about it being even or odd
Please tell me if I missed something.
"You don't have to believe in God, but you should believe in The Book." -Paul Erdős
Do the following problem:
f: R --> R with f(R)=Q (i.e. f takes on rational values) and f is continuous such that f(3)=10 then f(x) is
(C) increasing function
(D) nothing can be said about it being even or odd
since the function can be constructed in many ways like
f(x)=x(square) + 1, or,
f(x)=3x + 1, or,
like as you said
-10 for x ≤ -3
20 for -3 < x < 3
10 for x ≥ 3
so, it is not clear if any of the following condition holds true for all f(R)=Q
f(x) = f(-x) [for even function] or
f(-x) = -f(x) [for odd function] or
f(x) ≥ f(x'), given x ≥ x' [for increasing function]
Therefore nothing can be said about it being even or odd.
hey, @ sinistral i think constant functions are neither decreasing nor increasing, because for every x , u get the same f(x). pls see sir's defination of increasing function. and i too have a very stupid question- are all rational numbers even ? coz thats what i can recapitulate from ur ans.
Its given that a function is continuous and can take only rational values and f(3) =10
So, it must be the case that f(x) = 10 which is an even function.
Constant function is both increasing and decreasing.
A rational number is any number that can be expressed as a fraction p/q of two integers, with the denominator q not equal to zero. Since q may be equal to 1, every integer is a rational number.
So, we have odd rational numbers also.
Eg: 3, 5, 7, ....