Let $\mathcal{L}$ be a lenguage and $\mathcal{M}$ be a $\mathcal{L}$-structure.
prove that if $f: M^n \to M$ is definable then your image $f(M)$ is definable.
I know that if $f$ is definable then your graph $\mathcal{G}_f \subset M^{n+1}$ is definable, that is, there is a formule $\varphi(x_1, \cdots , x_n, x_{n+1}) $ such that, $$ g_1, \cdots , g_n, g_{n+1} \in \mathcal{G}_f \iff \mathcal{M} \models \varphi(g_1, \cdots , g_n, g_{n+1}) $$
but i don't know how to associate $\varphi$ with a formule that define to $f(M)$
Any help is important.
Thank your.
$$ y\in \operatorname{image}(f)\iff\exists \vec x\in M^n:(\vec x,y)\in G_f\iff \mathcal M\models \exists \vec x\varphi(\vec x,y)$$