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