16 mark for review the periodic function f has a period of 4. the function is increasing on the input…
16 mark for review the periodic function f has a period of 4. the function is increasing on the input - value interval (0,4). which of the following is true? a f(9) < f(15), because f(9)=f(1), f(3)=f(15), and f(1) < f(3). b f(9) > f(15), because the values are on two different increasing intervals. c f(9) < f(17), because 17 - 9 is a multiple of 4. d f(9) > f(17), because 17 - 9 is a multiple of 4.
Answer
Explanation:
Step1: Use the periodic - property of the function
If a function (f(x)) has a period (T = 4), then (f(x)=f(x + nT)) for any integer (n). For (f(9)), we can write (9=4\times2 + 1), so (f(9)=f(1)) (since (n = 2) and (x = 1)). For (f(15)), we can write (15=4\times3+3), so (f(15)=f(3)).
Step2: Analyze the increasing - property of the function
The function (f(x)) is increasing on the interval ((0,4)). That means if (a,b\in(0,4)) and (a < b), then (f(a)<f(b)). Since (1,3\in(0,4)) and (1 < 3), we have (f(1)<f(3)), which implies (f(9)<f(15)) (because (f(9)=f(1)) and (f(15)=f(3))).
Answer:
A. (f(9)<f(15)), because (f(9)=f(1)), (f(3)=f(15)), and (f(1)<f(3))