Question

(1) For all integers y, there is an integer x so that x^3 + x = y.

(2)For all integers x, x^3 + x is even

Thank you for your help.

Answer 1

(1) obviously false

x^3+x=2 has no integer solution.
x=0 is not a solution
if x>=1, so is x^3, so the sum is > 2.
and of course, if x<0, so is x^3, so their sum is negative.

(2) since x and x^3 are both odd, the sum is even.