Find whether f:Z - Z defined by f(x) = x² + 5 for all x belongs Z is one one or not.

the given function is $f:Z\to Z ; f\left(x\right)=x^2+5$ 
and $f(-x)=(-x{)}^2+5=x^2+5$
therefore $f\left(x\right)=f(-x)$
$$\begin{gathered} i.e. f\left(1\right)=f(-1) \\ f\left(2\right)=f(-2) \\ ... \\ ... \\ f\left(k\right)=f(-k) ; k\in Z \end{gathered}$$
therefore the function has same value for positive and negative x  thus function is not one-one.

hope this helps you.