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  • Which of the following functions from Z to Z are bijections? (a) f(x) = x3

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

\\(a) f(x) = x\textsuperscript{3 }\\(b) f(x) = x + 2 \\(c) f(x) = 2x + 1 \\(d) f(x) = x\textsuperscript{2} + 1 \\

Answers (1)

(b)

Given that f : Z \rightarrow Z \\

Say, x\textsubscript{1}, x\textsubscript{2} \in f(x) \rightarrow f(x\textsubscript{1}) = x\textsubscript{1}+ 2; f(x\textsubscript{2}) = x\textsubscript{2}+ 2;\\

For, f(x\textsubscript{1}) = f(x\textsubscript{2})\ \rightarrow x\textsubscript{1}+ 2 = x\textsubscript{2}+\ 2 \\

or, x\textsubscript{1 }= x\textsubscript{2}\\

Therefore, the function f(x) is one-one. 

Say, y = x + 2 

or, x = y - 2 \forall y \in Z \\

Hence, f(x) is onto. 

Therefore, the function f(x) is bijective.

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