An even function must always have what characteristic?

Answer: 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.