Show $\max\{f-g,0\} + g=\max\{f,g\}$

Question

As stated in the title:

Show that for a given variable $x$ it holds $max\{(f-g)(x),0\}+g(x)=max\{f(x),g(x)\}$, where $f,g$ are functions.

I just came across the above stated expression and I am wondering if it is sufficient to check the two cases $f(x)\geq g(x)$ and $f(x)<g(x)$. Or do I have to use/verify further assumptions?

Answer 1

How is $\max \{f,g\}$ defined?

Remember that for any two numbers $x,y$: $$\max \{x,y\} ={x+y+|x-y|\over 2}$$

and this is easy to prove.