$\displaystyle f$ and $\displaystyle g$ are functions from $\displaystyle R$ to $\displaystyle R$

1) If $\displaystyle f \circ g$ is a monotonically increasing function, does that mean that $\displaystyle f $is monotonically increasing function?

2) If $\displaystyle f \circ g$ and $\displaystyle f$ are monotonically increasing functions, does that mean that $\displaystyle g$ is a monotonically increasing function

I think for the first one $\displaystyle f$ must be a monotonically increasing funtion. It doesn't matter what $\displaystyle g$ is since it is in the domain of $\displaystyle f$. We can define $\displaystyle f$ however we want to but if $\displaystyle f \circ g$ is a monotonically increasing function, then definitely $\displaystyle f$ is too.

The second one I think is false for the reasons I just gave.

Am I correct?

Thanks!