Max of two functions notation

Question

I am working on a proof and the hypothesis included the following function:

$h(x)=$ max $\{f(x),g(x)\}$

I do not understand this notation or what this is saying. Could someone give me a formal definition and explain what "max" this function is referring to?

Answer 1

You may as well write $$ h(x)=\begin{cases}f(x)&\text{if }f(x)\ge g(x)\\g(x)&\text{otherwise}\end{cases}$$