"Defining Functions" discusses how you can define functions in the Wolfram Language. In a typical case, you would type in f[x_]=x^2 to define a function f. (Actually, the definitions in "Defining Functions" use the := operator, rather than the = one. "Immediate and Delayed Definitions" explains exactly when to use each of the := and = operators.)
The definition f[x_]=x^2 specifies that whenever the Wolfram Language encounters an expression that matches the pattern f[x_], it should replace the expression by x^2. Since the pattern f[x_] matches all expressions of the form f[anything], the definition applies to functions f with any "argument".
Function definitions like f[x_]=x^2 can be compared with definitions like f[a]=b for indexed variables discussed in "Making Definitions for Indexed Objects". The definition f[a]=b specifies that whenever the particular expression f[a] occurs, it is to be replaced by b. But the definition says nothing about expressions such as f[y], where f appears with another "index".
To define a "function", you need to specify values for expressions of the form f[x], where the argument x can be anything. You can do this by giving a definition for the pattern f[x_], where the pattern object x_ stands for any expression.
|f[x]=value||definition for a specific expression x|
|f[x_]=value||definition for any expression, referred to as x|
Making definitions for f or f[a] can be thought of as being like giving values to various elements of an "array" named f. Making a definition for f[x_] is like giving a value for a set of "array elements" with arbitrary "indices". In fact, you can actually think of any function as being like an array with an arbitrarily variable index.
In mathematical terms, you can think of f as a mapping. When you define values for, say, f and f, you specify the image of this mapping for various discrete points in its domain. Defining a value for f[x_] specifies the image of f on a continuum of points.
The Wolfram Language allows you to define transformation rules for any expression or pattern. You can mix definitions for specific expressions such as f or f[a] with definitions for patterns such as f[x_].
Many kinds of mathematical functions can be set up by mixing specific and general definitions in the Wolfram Language. As an example, consider the factorial function. This particular function is in fact built into the Wolfram Language (it is written n!). But you can use Wolfram Language definitions to set up the function for yourself.
The standard mathematical definition for the factorial function can be entered almost directly into the Wolfram Language, in the form f[n_]:=n f[n-1];f=1. This definition specifies that for any n, f[n] should be replaced by n f[n-1], except that when n is 1, f should simply be replaced by 1.