Question:
Which of the following functions from Z to Z are bijections?
(a) $f(x)=x^{3}$
(b) $f(x)=x+2$
(c) $f(x)=2 x+1$
(d) $f(x)=x^{2}+1$
Solution:
Given: $f: Z \rightarrow Z$
(a) $f(x)=x^{3}$
It is one-one but not onto.
Thus, it is not bijective.
(b) $f(x)=x+2$
It is one-one and onto.
Thus, it is bijective.
(c) $f(x)=2 x+1$
It is one-one but not onto.
Thus, it is not bijective.
(d) $f(x)=x^{2}+1$
It is neither one-one nor onto.
Thus, it is not bijective.
Hence, the correct option is (b).