If $f,g : \mathbb{R} \to \mathbb{R}$, how is max{$f$,$g$} defined? I am currently reading a textbook on metric spaces and came across the following terminology for two functions I can't seem to find anywhere how it is defined.
Let $f,g : \mathbb{R} \to \mathbb{R}$, how is max{$f$,$g$} defined? similarly, how is min{$f,g$} defined?

I am thinking for max{$f$,$g$}: this means simply to take the maximum values of $f$ and $g$ and max{$f$,$g$} consists of all those values. Similarly, to take the minimum values of $f$ and $g$ and min{$f,g$} consists of all those values.

The motivation for this is, I came across a problem where it asked: given two metrics $d_1$ and $d_2$ (for $(X_1,d_1)$ and $(X_2,d_2)$ respectively) is max{$d_1$,$d_2$} a metric on $X_1 \times X_2$? However, to begin to answer this question, I need to define the terminology I am unfamiliar with.

For each fixed $x$, $\max\{f(x),g(x)\}$ is the largest number between the two real numbers $f(x)$ and $g(x)$. Defining $\varphi(x)=\max\{f,g\}(x)=\max\{f(x),g(x)\}$ it is possible to prove that $ \varphi(x)=\frac{1}{2}[f(x)+g(x)+|f(x)-g(x)|]$.
