$f(x) + f(-x)$ is aways even

Question

I was having some troubles to help some friends with this question: Let $f:\mathbb{R} \to \mathbb{R}$ a function. Is true that $g(x) = f(x) + f(-x)$ is aways symmetric to the $y$ axis? If $f$ is even or odd, this is trivial, but if the function is neither even or odd? When it is a polynomial neither ever or odd, it's pretty easy to see that this is true, but there is some function continuous in $\mathbb{R}$ that this is not true? If doesn't, how to prove that it is true for every $f$? Thank you!

Answer 1

$$g(x)=f(x)+f(-x)$$

$$g(-x)=f(-x)+f(x)$$