CAT 2004 Question 4

Instructions

Directions for the following two questions:

Answer the questions on the basis of the information given below.

$$f_1(x) = x$$ if $$0 \leq x \leq 1$$ $$f_1(x) = 1$$ if x >= 1 $$f_1(x) = 0$$ 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 x

Question 4

How many of the following products are necessarily zero for every x:$$f_1(x)f_2(x), f_2(x)f_3(x), f_2(x)f_4(x)$$

Solution

Checking for different values of x . Suppose x= -0.5 we get

$$f_1(x)f_2(x) = 0*0.5 = 0$$

$$f_2(x)f_4(x) = 0.5*0 = 0$$ .

But $$f_2(x)f_3(x)$$ is not equal to zero.

Hence two functions are necessarily equal to zero and two products given above are equal to zero.

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