# Confusing fact about minimum and maximum

Let $f(x)$ and $g(x)$ be functions. I'm looking for an upper bound on $-\min(-f(x), -g(x))$, but the minus signs are making me dizzy. Is this the same as an upper bound on $\max(f(x),g(x))$?

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Is this easier to see: $\min(-a,-b)=-\max(a,b)$? –  Srivatsan Sep 23 '11 at 14:24

Pick a number $K$.
$$-\operatorname{min}(-f,-g) \leq K \iff \operatorname{min}(-f,-g) \geq -K$$ Using Srivatsan's suggestion, this is $$-\operatorname{max}(f,g) \geq -K$$ which is equivalent to $\operatorname{max}(f,g) \leq K$. So you were right!