Is f(x) An Odd Integer?

[nycmath]

Precalculus
Michael Sullivan
Edition 10
Chapter 3, Section 3.3

See attachment

We are officially done with Section 3.3.

[Eigenvalue]

To find if f(x) is an odd integer, we can use "x" as a placeholder value for odd integers and "2x" as a placeholder value for even integers--because even integers are divisible by 2

f(x)=ax²+bx+c
As a, b, and c are all odd integers, f(x) is odd; the sum of three odd integers is odd

f(2x)=a(2x)²+b(2x)+c
=4ax²+2bx+c
As a, b, and c are all odd integers, f(x) is odd; the sum of two even integers and an odd integer is odd

The outputs for f(x) are always odd.