Difference between Map[f[#] + g[#] &, {a, b, c}] and Map[f[x]+g[x]&, {a, b, c}]

Question

  Map[f[#] + g[#] &, {a, b, c}] vs Map[f[x]+g[x]&, {a, b, c}]

My question is: why is the output different?

Map[f[#] + g[#] &, {a, b, c}]

(* {f[a] + g[a], f[b] + g[b], f[c] + g[c]} *)

and

Map[f[x] + g[x] &, {a, b, c}]

(* {f[x] + g[x], f[x] + g[x], f[x] + g[x]} *)

Answer 1

The Map function lets you apply a function to every element of a list. So:

Map[Sin, {0, Pi/2, Pi}]

applies Sin to each of the three values in the list which occupies the second place in the Map function call. The result is a list of numbers:

{0, 1, 0}

If you define your own functions, you can use them in a similar way:

double[x_] := x + x;

Map[double, {0, Pi/2, Pi}]

{0, π, 2 π}

You can use # or Slot, together with the ampersand (&), to refer to the arguments or parameters of a function, like this:

Map[Sin[#] + Cos[#] &, {0, Pi/2, Pi}]

{1, 1, -1}

When Mathematica doesn't understand what you typed, it returns it to you unchanged:

Map[f [x] g[x] &, {0, Pi/2, Pi}] 

{f[x] g[x], f[x] g[x], f[x] g[x]}

The syntax highlighter in the notebook should indicate which parts of your code don't mean anything to Mathematica. On my machine, it shows in blue: