I would like to know the correct notation of inverse function because I want to show that $min(f(x)) \equiv max(f^{-1}(x))$. My colleague corrects me that I should write $min(f(x)) \equiv max(-f(x))$.
Can you please tell my which one is correct? or both? Than you for your answer and explanation.
Best regards,
Both are incorrect.
Counterexample to the first one : $f(x)=x^2$ on $\mathbb{R}_+^*$.
One has $\min_{x \geq 0} f(x) = 0$, but $\max_{x \geq 0} f^{-1}(x)=\max_{x \geq 0} \sqrt{x}$ is undefined.
Counterexample to the second one : $f(x)=x$ on $[1,2]$.
One has $\min_{1 \leq x \leq 2} f(x)=1$, but $\max_{1 \leq x \leq 2} -f(x) = -1$.