# Math Help - Interesting Question

1. ## Interesting Question

A function f is known to have the following properties for all t: f(t+4)=f(t) and f(2+t)=f(2-t). Which of the following must be true?

A) f is periodic with period 4
B) f(t)>0 for all t
C) f(0)=0
D) f is an even function
E) f is an odd function

A function f is known to have the following properties for all t: f(t+4)=f(t) and f(2+t)=f(2-t). Which of the following must be true?

A) f is periodic with period 4
B) f(t)>0 for all t
C) f(0)=0
D) f is an even function
E) f is an odd function
A) is true: f(t + 4) = f(t).

D) is true: Let t --> t + 2 in f(2+t)=f(2-t): f(t + 4) = f(-t). But f(t + 4) = f(t). Therefore f(-t) = f(t) and so f is even.

C) is not necessarily true. It's easy to construct examples that satisfy A) and D) but not C).