Let $f:R^3→R^3$ diffeomorphism. Prove that for every smooth surface $M \subset R^3$ the set $f(M)$ is also a smooth surface. So to prove that $f(M)$is a smooth surface I need to find a smooth parametrization (1-1 differentiable onto with continuous inverse) $ψ:W⊂R^2→f(M)$ for $f(M)$ to be surface. So I have something like this ($φ$ is a smooth parametrization since I know M is a smooth surface).
$$\begin{array} MM&\stackrel{f}{\longrightarrow}&f(M)\\ \uparrow{φ}&&\uparrow{ψ=?}\\ U\subset R{^2}&\stackrel{??}{\longrightarrow}&W\subset R{^2}\ \end{array}$$
I need to find $ψ$. Will $ψ=φ \circ f$ work?
The restriction of a diffeomorphism to an open submanifold is a diffeomorphism onto its image. So your composition should work. That is, if $\lbrace \varphi_\lambda\rbrace_{\lambda\in\Lambda}$ is a collection of charts for $M$ then $\lbrace f\circ \varphi_\lambda\rbrace_{\lambda\in\Lambda}$ is a collection of charts for $f(M)$.