Question

Determine whether each of these functions from the set of integers to the set of integers is injective, surjective, or bijective. f(x)=1+X^2 f(x)=2x f(x)=17+x

Answer 1

Given that f: Z → Z

Injective: For every element in domain should have a uniquee and only one image.

Surjective: Every element in co-domain should be mapped by atleast one elemenet in domain.

Bijective : If a function is both injective and bijective, then it is called as bijective.

Question 1) f(x) = 1+x^2

f(x) is not injective, since f(-1) = 1+1=2, f(1)=1+1=2, So there is no unique image

f(x) is not surjective, since -1 in co-domain do not mapped any element domain

f(x) is not bijective, since it is neither injective nor surjective.

Question 2) f(x) = 2x

f(x) is injective, since f(-1) = 2*-1=-2, f(1)=2*1=2, So there is unique image for every element in domain

f(x) is not surjective, since -1 in co-domain do not mapped any element domain. To get -1 x should be -(1/2), but it is not integer

f(x) is not bijective, since it injective but not surjective.

Question 3) f(x) = 17+x

f(x) is injective, since f(1) = 17+1=18, f(2)=17+2=19, So there is unique image for every element in domain

f(x) is urjective, since every element of co-domain can be mapped by atleast one-elemnt od domain

Ex: To get -1 x should be -18, to get 1 x should be -16

f(x) is bijective, since it injective and surjective.