To make $K(x, x')$ a valid kernel function, it needs to satisfy the following conditions:
Symmetry: \( K(x, x') = K(x', x) \) for all inputs \( x \) and \( x' \).
Non-negativity: \( K(x, x) \geq 0 \) for all inputs \( x \).
The original function $K(x, x') = f(x)g(x') + f(x')g(x)$ lacks symmetry. Adding the terms
$f(x)g(x)$ and $f(x')g(x')$ addresses this issue:
New function: $K(x, x') = f(x)g(x') + f(x')g(x) + f(x)g(x) + f(x')g(x')$
This new function satisfies both conditions:
Symmetry: It is evident that \( K(x, x') = K(x', x) \) due to the arrangement of terms.
Non-negativity:It's always true that \( f(x)g(x) \geq 0 \) and \( f(x')g(x') \geq 0 \) for
real-valued functions \( f \) and \( g \), ensuring non-negativity.
Therefore, by adding the terms B) $f(x)g(x) + f(x')g(x')$ , we create a valid kernel function.