Let $\text{}f\text{}$ be any function which is defined for all numbers. Show that $\text{}g(x)=f(x)+f(-x)\text{}$ is even.

$\begin{array}{}\text{(1)}& \mathrm{e}.\mathrm{g}.\text{}f(x)={x}^{2}\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}g(x)={x}^{2}+(-x{)}^{2}=2{x}^{2}\leftarrow \text{}\text{}\text{even}\end{array}$

$\begin{array}{}\text{(2)}& \mathrm{e}.\mathrm{g}.\text{}f(x)={x}^{3}\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}g(x)={x}^{3}+(-x{)}^{3}=0\leftarrow \text{}\text{}\text{even}\end{array}$

$\begin{array}{}\text{(3)}& \mathrm{e}.\mathrm{g}.\text{}f(x)={x}^{3}+{x}^{2}\text{}\text{}\leftarrow \text{}\text{}\text{neither even nor odd}\end{array}$

$\begin{array}{}\text{(4)}& \phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}\text{}g(x)=({x}^{3}+{x}^{2})+(-{x}^{3}+{x}^{2})=2{x}^{2}\leftarrow \text{even}\end{array}$

But how it can be proven that the claim holds?

And needless to say, can I completely assume "for all numbers" in the problem statements belong to a set of complex numbers(handling imaginary numbers is also required in this problem)?

$\begin{array}{}\text{(1)}& \mathrm{e}.\mathrm{g}.\text{}f(x)={x}^{2}\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}g(x)={x}^{2}+(-x{)}^{2}=2{x}^{2}\leftarrow \text{}\text{}\text{even}\end{array}$

$\begin{array}{}\text{(2)}& \mathrm{e}.\mathrm{g}.\text{}f(x)={x}^{3}\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}g(x)={x}^{3}+(-x{)}^{3}=0\leftarrow \text{}\text{}\text{even}\end{array}$

$\begin{array}{}\text{(3)}& \mathrm{e}.\mathrm{g}.\text{}f(x)={x}^{3}+{x}^{2}\text{}\text{}\leftarrow \text{}\text{}\text{neither even nor odd}\end{array}$

$\begin{array}{}\text{(4)}& \phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}\text{}g(x)=({x}^{3}+{x}^{2})+(-{x}^{3}+{x}^{2})=2{x}^{2}\leftarrow \text{even}\end{array}$

But how it can be proven that the claim holds?

And needless to say, can I completely assume "for all numbers" in the problem statements belong to a set of complex numbers(handling imaginary numbers is also required in this problem)?