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Question
Which of the following functions from Z into Z are bijections?
Options
f(x) = x3
f(x) = x + 2
f(x) = 2x + 1
f(x) = x2 + 1
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Solution
f(x) = x + 2
Explanation:
For bijection on Z, f(x) must be one-one and onto
Function f(x) = x2 + 1 is many-one as f(1) = f(–1)
Range of f(x) = x3 is not Z for x ∈ Z.
Also f(x) = 2x + 1 takes only values of types = 2k + 1 for x = k ∈ Z
But f(x) = x + 2 takes all integral values for x ∈ Z
Hence f(x)= x + 2 is bijection on Z
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