An even function must always have what characteristic?

  If the function f(x) is even, then f(x) = f(-x).

Examples:
1. f(x) = x² + 3
f(-x) = (-x)² + 3
       = x² + 3
       = f(x)
Therefore f(x) is even.

2. f(x) = 3x⁶ - 4x⁴ + 5x² + 1
f(-x) = 3(-x)⁶ - 4(-x)⁴ + 5(-x)² + 1
       = 3x⁶ - 4x⁴ + 5x² + 1
       = f(x)
Therefore f(x) is even.