Suppose $g: \mathbb R^n \to [0,\infty]$ and $h: \mathbb R^n \to [0,\infty)$. Let $f = g-h$. For any function $j: \mathbb R^n \to [0,\infty]$, let $j_M(x) = \min\{j(x),M\}$ for $M \in \mathbb N$.
Is it true that $f_M = g_M - h_M$?
Attempt: Let $x \in \mathbb R^n$. If $f_M(x) = M$, then $g(x) \geq M + h(x) \geq M$, so $g_M(x) = M$. But there's no reason why $h(x)=0$, so it doesn't look like the result will hold. Is this the right idea?
No. Consider $g(x)=2x, h(x)=x$. Then $f(x)=x$. If $x=\frac{3M}{4}$, then $g_M(x)=\min(\frac{3M}{2},M)=M, f_M(x)=h_M(x)= \frac{3M}{4}$. So $$ f_M(x)=\frac{3M}{4} > \frac{M}{4}= g_M(x)-h_M(x)$$