Answer to Question #131445 in Discrete Mathematics for jaya

Question #131445
Draw the graph of the function f(n) = 1 - n
2
from the set of integers to set of integers.
1
Expert's answer
2020-09-08T15:13:19-0400

The function f(n)=1n2,nZf(n)=1-n^2, n\in\Z is not one-to-one:

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

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

Hence f(1)=0=f(1),f(-1)=0=f(1), but (11)(-1\not=1)

The function f(n)=1n2,nZf(n)=1-n^2, n\in\Z is not onto:

there is no integer nn with f(n)=1n2=2,f(n)=1-n^2=2, for instance.

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


f(n)=1(n)2=1n2=f(n),nZf(-n)=1-(-n)^2=1-n^2=f(n), n\in\Z

The function f(n)=1n2,nZf(n)=1-n^2, n\in\Z is even.





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