Re: functions problem

Posted by Amit Goyal on May 02, 2013; 3:18am
URL: http://discussion-forum.276.s1.nabble.com/functions-problem-tp7580283p7580311.html

Problem:
f: R to Q such that f(3)=10 then f(x) is
(A) odd
(B) even
(C) increasing function
(D) nothing can be said about it being even or odd

Definitions: 
Let f(x) be a real-valued function of a real variable. Then f is even if the following equation holds for all x and -x in the domain of f:
f(x) = f(-x)

Let f(x) be a real-valued function of a real variable. Then f is odd if the following equation holds for all x and -x in the domain of f:
-f(x) = f(-x)

Let f(x) be a real-valued function of a real variable. Then f is increasing if the following holds for all x and x' in the domain of f:
x ≥ x'  implies f(x) ≥ f(x')

Solution:
Consider the following f: R to Q
f(x) =
-10 for x ≤ -3
20 for -3 < x < 3
10 for x ≥ 3

Note that f satisfies f(3) = 10,
f is not increasing because f(2) > f(3),
f is not odd because f(-2) ≠ -f(2),
f is not even because f(-3) ≠ f(3).
Thus, we rule out options (A), (B) and (C).