Q. 145.0( 1 Vote )

Mark the correct alternative in each of the following:

Which of the following functions from Z to itself are bijections?

A. f(x) = x3

B. f(x) = x + 2

C. f(x) = 2x + 1

D. f(x) = x2 + x

Answer :

a. f(x) = x3


For no value of x ϵ Z, f(x) = 2.


Hence, it is not bijection.


b. f(x) = x + 2


If f(x) = f(y)


x + 2 = y + 2


x = y


So, f is one-one.


Also, y = x + 2


x = y – 2 ϵ Z


So, f is onto.


Hence, this function is bijection.


c. f(x) = 2x + 1


If f(x) = f(y)


2x + 1 = 2y + 1


x = y


So, f is one-one.


Also, y = 2x + 1


2x = y – 1



So, f is into because x can never be odd for any value of y.


d. f(x) = x2 + x


For this function if we take x = 2,


f(x) = 4 + 2


f(x) = 6


For this function if we take x = -2,


f(x) = 4 - 2


f(x) = 2


So, in general for every negative x, f(x) will be always 0. There is no x ϵ R for which f(x) ϵ (-∞, 0).


It is not bijection.

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