Based on the graph provided, we can discern that it is very much like the cosine graph, following the same curvature and period as the function f(x) = cos(x).
Expanding on given information, such that the Y intercept is at (0,4) as well as alternating and recurring points of (-π + 2nπ, -4) and (π + 2nπ, 4), with a maximum and minimum of 5, it is easy to see that it is a x-shifted cosine graph with an amplitude of 5.
Given this, we can find that this graph satisfies the function:
f(x-cos^-1(4/5))=5*cos(x)
This result was achieved by initially letting f(x-z) = 5*cos(x), and beginning to look towards a mathematical approach to finding the offset. The approach I used (for ease and effectiveness) was to solve for z when 5*cos(x-z) = 4 while x = 0. This led me to find that z = cos^-1(4/5).
The question 'Do we always get new functions from old ones?', is difficult to answer, sometimes we perceive that we have a new function from an old one, an example being the quadratic formula's relationship with f(x)=ax^2+bx+c, where we acknowledge the former being a separate one from the latter, where it is just some 'rearrangement' of the same function. Then in our example where the function was some transformation of the cosine function, we are tempted to say that it is not a new function, just some 'changing' of the original one, where widely accepted information shows that it is.
My understanding is that every function that is 'transformed' from an old one, is not a new one, but simply a different version of it, what that new function does could be similar or different from its original, but nonetheless, the same.