Answer the following questions on the basis of the information given below:
f1(x)= x for 0 ≤ x ≤ 1
= 1 for x = 1
= 0 otherwise
f2(x)= f1(-x) for all x
f3(x)= -f2(x) for all x
f4(x)= f3(-x) for all x
Q1) How many of the following products are necessarily zero for every x
f1(x)f2(x), f2(x)f3(x), f2(x)f4(x)?
(1)0 (2)1
(3)2 (4)3
Q2) Which of the following is necessarily true?
(1)f4(x) = f1(x) for all
x (2)f1(x) =
-f3(-x) for all x
(3)f2(-x) = f4(x) for all
x (4)f1 (x) + f3(x)
= 0 for all x
To enter your answers, click on comments below:
koi to batao
ReplyDeleteWe have to consider positive and negative numbers for all the cases.
ReplyDeletef1(x) is positive for positive numbers, and 0 for negative numbers. (0 for x = 0)
f2(x) is 0 for positive numbers, and positive for negative numbers. (0 for x = 0)
f3(x) is 0 for positive numbers, and negative for negative numbers. (0 for x = 0)
f4(x) is negative for positive numbers, and 0 for negative numbers. (0 for x = 0)
So, we see that, out of the 3 products in the question, f1(x)*f2(x) and f2(x)*f4(x) are always zero, for any x.
Second sub question,f4(x) = f3(-x) = -f2(-x) = -f1(x). Hence, 1st option is false.
–f3(-x) = f2(-x) = f1(x). Hence this is true.