I've got functions $\displaystyle f,g:\mathbb{R}\rightarrow \mathbb{R} $ . $\displaystyle f $ is strictly increasing and $\displaystyle f\circ g $ is strictly decreasing I am asked to show $\displaystyle g $ is strictly decreasing. I already have a solution in front of me however I would like to know if this approach is correct.

Suppose $\displaystyle x_{1} < x_{2} $ then $\displaystyle f\left ( g\left ( x_{1} \right ) \right ) > f\left ( g\left ( x_{2} \right ) \right )$ $\displaystyle \Rightarrow g\left ( x_{1} \right ) > g\left ( x_{2} \right ) $ since $\displaystyle f $ is strictly increasing.

therefore $\displaystyle g $ is strictly decreasing.