So, I have this function:
f[z_] = Exp[-z] Sin {2 Pi [0.61 z - 0.22z^2 +2 g[z_]};
and this function
g[z_]=Cos[2 Pi (1.5 z - z^2 )];
How do I write one inside the other with all the parentheses, braces and square brackets in the right place to be able to write an algorithm?
I started testing Mathematica now and I’ve been trying for days to understand what I’ve been doing wrong, because the answer doesn’t return the sine and cosine by mistake in the square brackets, parentheses…
The underscore z_ indicates a pattern, and belongs to the left-hand side. Square brackets are for the argument of functions, curly braces for lists, round parentheses to indicate the order of operations. It is VERY coherent and worth internalizing the distinction, which is lost to traditional math notation:
g[z_] := Cos[2 Pi (1.5 z - z^2)];
f[z_] := Exp[-z] Sin[2 Pi (0.61 z - 0.22 z^2 + 2 g[z])];
Plot[f[z], {z, 0, 2}]
Check again your syntax. The argument of functions like Sin and Cos goes inside square brackets. Pi is not a function, it is a numerical constant, it is not followed by [].
Ok… How would I write all in one function?
f[z_] := Exp[-z] Sin [2 Pi (0.61 z - 0.22 z^2 + 2 (Cos 2 Pi [1.5 z - z^2]))]
?
I would really appreciate if you could write this down, once I know how to do it I won’t go wrong again… You would help a lost student! I’m reading what you’re writing, but I have a lot of information for a function and I don’t know how to manage it! All the parentheses and square brackets with the pi, the sine and the cosine are making fun of me! 
2Pi is multiplied by 1.5 z - z^2, and this is indicated by either explicitly by 2 Pi*(1.5 z - z^2) or implicitly by juxtaposition 2 Pi(1.5 z - z^2). The result is the argument of Cos, and goes within square brackets: Cos[2 Pi (1.5 z - z^2)]. Altogether
f[z_] :=
Exp[-z] Sin[2 Pi (0.61 z - 0.22 z^2 + 2 Cos[2 Pi (1.5 z - z^2)])]