Answer to Question #131445 in Discrete Mathematics for jaya

Question

Draw the graph of the function f(n) = 1 - n
2
from the set of integers to set of integers.

Answer 1

The function $f(n)=1-n^2, n\in\Z$ is not one-to-one:

$f(-1)=1-(-1)^2=0$

$f(1)=1-(1)^2=0$

Hence $f(-1)=0=f(1),$ but $(-1\not=1)$

The function $f(n)=1-n^2, n\in\Z$ is not onto:

there is no integer $n$ with $f(n)=1-n^2=2,$ for instance.

If $n\in\Z,$ then $-n\in\Z$ and

$$f(-n)=1-(-n)^2=1-n^2=f(n), n\in\Z$$

The function $f(n)=1-n^2, n\in\Z$ is even.