Symmetric and Periodic

Starting with $6 = 8 - 2,$ we see that $f(6) = f(8-2) = f(-2 + 8 ) = f(-2)$ . The last equality holds because the function has period 8. Since the function is even ( $f(-x) = f(x)$ ), $f(6) = f(-2) = f(2) = 3,$ and so $f(2) + f(6) = 3 + 3 = 6.$

Since $f(x)$ has a period of 8, $f(x) = f(x+8n)$ , where $n$ is an integer. Then $f(2) = f(2-8) = f(-6)$ . Now $f(x)$ is also even. This means that $f(x)=f(-x)$ . Therefore, $f(2) = f(-6) = f(6) = 3$ and $f(2) + f(6) = 3+3=\boxed{6}$ .