My understanding is that the transformation $f(ax)$ to a transformation $f(x)$ will increase the frequency of $f(x)$ by scale factor $a$. Furthermore, $a$ can itself be a function of $x$.
Thus if we consider $f(x)=(\sin x)^2$ and $a=x^{−1/2}$, this is how the curve looks like this : https://i.stack.imgur.com/i4aCx.png
But shouldn’t the frequency of this curve be very high for small values of $x$ (since a gets very large for small values of $x$)?
Please can someone explain/clarify?
Many thanks
Your understanding is correct, to a point. The point at which it stops being correct is
A scalar is simply an element of some algebraic structure (usually a ring or a field), whereas a function is quite a different thing. Formally speaking, a function is a subset of the Cartesian product of its domain and codomain — in other words, it's a set of ordered pairs (or triples if it's in two variables, etc) which certainly do not behave like scalars (at least, they're not required to). If $a = x^{1/2}$, then $f(ax) = f(x^{1/2}x) = f(x^{3/2})$, which is a composition of functions.