Answer the following questions on the basis of the information given below:

f

_{1}(x)= x for 0 ≤ x ≤ 1
= 1 for x = 1

= 0 otherwise

f

_{2}(x)= f_{1}(-x) for all x
f

_{3}(x)= -f_{2}(x) for all x
f

_{4}(x)= f_{3}(-x) for all xQ1) How many of the following products are necessarily zero for every x

f

_{1}(x)f

_{2}(x), f

_{2}(x)f

_{3}(x), f

_{2}(x)f

_{4}(x)?

(1)0 (2)1
(3)2 (4)3

Q2) Which of the following is necessarily true?

(1)f

_{4}(x) = f_{1}(x) for all x (2)f_{1}(x) = -f_{3}(-x) for all x
(3)f

To enter your answers, click on comments below:_{2}(-x) = f_{4}(x) for all x (4)f_{1}(x) + f_{3}(x) = 0 for all x
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.