Because $f(x)$ is even and $g(x)$ is odd, therefore $$f(-x)+g(-x)=f(x)-g(x)=\sqrt{1-x+x^2}$$ Adding this equation to the given relation yields $$f(x)=\frac{1}{2}\big(\sqrt{1+x+x^2} + \sqrt{1-x+x^2}\big)$$ Therefore $$f(3)=\frac{1}{2}\big(\sqrt{1+3+3^2} + \sqrt{1-3+3^2}\big)=\boxed{\frac{\sqrt{13}+\sqrt{7}}{2}}$$