Describing transformations of functions

I was doing a question which asked to describe in words the set of transformations to get from the graph of y = √x - x/4 to the graph of y = √(4x+1) - x - 1.

I wrote the answer in words but I wondered if there is a nice way to describe this using maths notation. E.g. I could say this

Let f(x) = √x - x/4
f(4x) = √(4x) - (4x)/4
Let g(x) = f(4x)
g(x+1/4) = √(4x+1) - (4x+1)/4
Let h(x) = g(x+1/4)
h(x-3/4) = √(4x+1) - x - 1

Can this process be written in a simpler way, without having to define three functions?

If f is the original graph, then f(4x+1)-3/4 is the 2nd graph. But it's not really any simpler - it's just removing all intermediate steps/functions.

This is incorrect.

Your $h(x-3/4)$ should read as $\sqrt{4x+1} - x + \dfrac{1}{2}$. I think you mean $h(x) - 3/4$.

Define $f(x) = \sqrt{x} - \dfrac{x}{4}$ and $g(x) = \sqrt{4x+1} - x - 1$.

Then $g(x) = f(4x+1) - \dfrac{3}{4}$.

These are 3 transformations; one stretch horizontally, one tanslation horizontally, one translation vertically.