How to find the max of two functions?

Question

Suppose I have a parametric function like f(v)=max{v-x,-y} where 0<=v<v_max, and x>=0, y>=0. I want to find simplify f(v) as follows.

f(v)=max{v-x,y}= -y if v<x-y, v-x if v>=x-y
I use the following code, but the problem is that if x-y<0, then the condition -y if v<x-y will not be true.

f[v_, x_] := v - x;
g[y_] := -y;
h[v_, x_, y_] :=  f[v, x]*Boole[f[v, x] >= g[y]] + g[y] (1 - Boole[f[v, x] >= g[y]]) // Simplify;

Is it possible to explicitly have the conditions that are not listed in True?

simplifying expressions

Roman · Answer

Would an explicit Piecewise do what you need?
h[v_, x_, y_] = 
  Piecewise[{{f[v, x], f[v, x] >= g[y]},
             {g[y],    f[v, x] < g[y]}}]

$$
left{
begin{array}{ll}
v-x & text{if }v-xge -y
-y & text{if }v-x< -y
0 & text{if True}
end{array}
right.
$$
Please note the immediate assignment. It makes little sense to have a Simplify statement in a delayed assignment.

Bob Hanlon · Answer

Clear["Global`*"]

f[v_, x_] := v - x;
g[y_] := -y;
h[v_, x_, y_] := 
  f[v, x]*Boole[f[v, x] >= g[y]] + g[y] (1 - Boole[f[v, x] >= g[y]]) // 
   Simplify;

h[v, x, y]

The condition represented by True is
Not@h[v, x, y][[1, 1, -1]]

(* v + y < x *)

For multiple conditions,
integration = 
 Assuming[vmax > 0, Integrate[h[v, x, y], {v, 0, vmax}] // Simplify]

The condition represented by True is
ConditionalExpression[Simplify@Not[Or @@ integration[[1, All, -1]]], 
 vmax > 0]

How to find the max of two functions?

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