Is the composition of a monotonic/strictly increasing/decreasing function $f$ with another function $g$ also monotonic/strictly increasing/decreasing? I believe, I read a theorem that stated that if $f(x)$ is a monotonic increasing/decreasing or strictly increasing/decreasing then $g(f(x))$ is also monotonic increasing/decreasing or strictly increasing/decreasing. Is this true?

It is not true in general since it depends also from $g(x)$.

Let consider for example

* $f(x)=x$

* $g(x)=1$




More in general for $f(x)$ strictly increasing we have

* $g(x)$ strictly increasing $\implies g(f(x))$ strictly increasing
* $g(x)$ monotonic increasing $\implies g(f(x))$ monotonic increasing
* $g(x)$ strictly decreasing $\implies g(f(x))$ strictly decreasing
* $g(x)$ monotonic decreasing $\implies g(f(x))$ monotonic decreasing



but for $f(x)$ monotonic increasing we have

* $g(x)$ strictly increasing $\implies g(f(x))$ monotonic increasing
* $g(x)$ monotonic increasing $\implies g(f(x))$ monotonic increasing
* $g(x)$ strictly decreasing $\implies g(f(x))$ monotonic decreasing
* $g(x)$ monotonic decreasing $\implies g(f(x))$ monotonic decreasing



and so on.