# Is the composition of a measurable function with a monotone function measurable?

Assume that $\phi:\mathbb{R}\rightarrow\mathbb{R}$ is a strictly monotonically increasing function. Is it true that a real valued function $f:X\rightarrow\mathbb{R}$ is measurable on $(X,M)$ iff its composition $\phi \circ f$ with $\phi$ is measurable.

What if $\phi$ is just a monotone function?

Hint for first part: $f$ is measurable if $f^{-1}([a,\infty)) \in M$ for all $a \in \mathbb R$.
Hint for second part: what if $\phi$ is constant?