When we substitute variables do we compose functions? Suppose that we have the function $f(u)=2u+3$ and we define $u=2x$. What of the below expressions are correct?
$$f(u(x))=4x+3 \ \text{or} \ f(x)=4x+3 $$
Does the "definition" serves only as a substitution (shorthand) i.e. is "$f$" still the same function or the composition of $f \circ u$ irrespective of the fact that we didn't write $u(x)=2x$ at first (abuse of notation)?

The second of your suggestions is the correct interpretation. We often abuse notation and write $f(x)$, although one actually defines a new function by composition $\hat f(x)=f\circ u(x)$. The most common application is the substitution, when calculating integrals. The error cancels in the end of those calculations by resubstitution (for indefinite integrals) or because you also substitute the boundaries (for definite integrals).
