Answer
Proved that $f(x)$ is strictly decreasing if and only if $g(x)=1/f(x)$ is strictly increasing.
Work Step by Step
Let $f:R\to R$ and $f(x)>0$ for all $x\in R$. Let $g(x)=1/f(x)$.
Assume that $f(x)$ is strictly decreasing.
Then, $xf(y)$
$\Rightarrow 1/f(x)<1/f(y)$ (Since $f(x)>0$ for all $x$)
$\Rightarrow g(x) f(y)$
$\Rightarrow$ f is strictly decreasing.
Thus, $f(x)$ is strictly decreasing if and only if $g(x)=1/f(x)$ is strictly increasing.
