Pictures in introductory texts to Morse theory are often drawn as to interpret a Morse function as a height function. Typically, an embedding of a torus into $\mathbb{R}^3$ is drawn, and the Morse function is then the height function by projecting onto one component (call the projection $\pi$).

This is a great picture because I have the feeling that any embedding can be perturbed to give a Morse function that way. Such a construction is used - as far as I can tell - in Lurie's definition of $(\infty,1)$-categories of bordisms. But is this a good picture for *any* Morse function?

In formulas, is every Morse function $f: M \to \mathbb{R}$ of following the form: $$f: M \hookrightarrow \mathbb{R}^n \stackrel{\pi}{\to} \mathbb{R}$$