Does adding a constant preserve argmax?

Question

Is $argmax (f(x)) = argmax (f(x) + c)$?

This property is not listed in any of the argmax properties. Intuitively this seems to be true as adding constant just shifts the graph upwards in the coordinate plane. This shouldn't change the corresponding 'x' value.

Answer 1

Yes, adding constant preserve it, suppose $x_1 \in \arg\max(f(x))$, then $$f(x_1) \ge f(x), \forall x \in X $$ hence

$$f(x_1)+c \ge f(x)+c, \forall x \in X $$

Hence $x_1 \in \arg\max (f(x)+c)$

the reverse direction is similar, we can subtract $c$ from both sides.