Let us consider the following function $f:X\rightarrow{R}$. Let us denote by $f_{max}$ and $f_{min}$, respectively, maximum and minimum values of $f$ function. Let us consider the following monotonic function $g:X\rightarrow{R}$. By using monotonic transformation of $f$ we obtain the following function $h:X\rightarrow{R}$, where $h=g(f)$. Let us denote by $h_{max}$ and $h_{min}$, respectively, maximum and minimum values of $h$ function.
What is the relationship between $f_{max}$ and $h_{max}$ and $f_{min}$ and $h_{min}$?
P.S. I know that for monotonic transformation the extreme values match. In what cases it is true?