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Question
Classify the following function as injection, surjection or bijection : f : Z → Z given by f(x) = x3
Solution
f : Z → Z, given by f(x) = x3
Injection test:
Let x and y be any two elements in the domain (Z), such that f(x) = f(y)
f(x) = f(y)
x3 = y3
x = y
So, f is an injection.
Surjection test :
Let y be any element in the co-domain (Z), such that f(x) = y for some element x in Z(domain).
f(x) = y
x3 = y
x = 3sqrty which may not be in Z.
For example, if y = 3,
x =`3 sqrt3` is not in Z .
So, f is not a surjection and f is not a bijection .
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