Addition of Functions

Question

So generally, if you have two functions f,g: X -->Y, and if there is some binary operation + defined on Y, then f + g has a canonical definition as the function x --> f(x) + g(x).

What's the best way to implement this in Mathematica?

f[x_] := x^2
g[x_] := 2*x
h = f + g;
h[1]

yields

(f + g)[1]

as an output

of course,

H = Function[z, f[z] + g[z]];
H[1]

Yields '3'.

Answer 1

Consider:

In[1]:= Through[(f + g)[1]]

Out[1]= f[1] + g[1]

To elaborate, you can define h like this:

h = Through[ (f + g)[#] ] &;

If you have a limited number of functions and operands, then UpSet as recommended by yoda is surely syntactically cleaner. However, Through is more general. Without any new definitions involving Times or h, one can easily do:

i = Through[ (h * f * g)[#] ] &
i[7]

43218

Answer 2

Another way of doing what you're trying to do is using UpSetDelayed.

f[x_] := x^2;
g[x_] := 2*x;

f + g ^:= f[#] + g[#] &; (*define upvalues for the operation f+g*)

h[x_] = f + g;

h[z]

Out[1]= 2 z + z^2

Also see this very nice answer by rcollyer (and also the ones by Leonid & Verbeia) for more on UpValues and when to use them