4.5 Defining Mathematica Functions
Mathematica TE contains hundreds of built-in mathematical functions like Round, Sqrt, and Sin, and these can be combined to form an endless number of possible expressions. These expressions can be simple like 1/x or more complicated like Abs[Abs[x] - 1]. The first step in Mathematica TE programming is to define a function to abbreviate such an expression. Part 3 describes Mathematica TE's powerful programming language in detail.
This defines the function f. The _ (" blank") after the 
means that the function's variable is 
.
In[1]:= f[x_] := 1 / (1 + x^2)
You can evaluate f for any 
you want, without retyping the expression.
In[2]:= f[2]
Out[2]= 
Suppose you give 
a value.
In[3]:= x = 10
Out[3]= 
Evaluating the function doesn't affect the value of 
.
In[4]:= {f[2/3.], x}
Out[4]= 
You can use any argument for f, and you can combine f with any other function, whether built-in or defined.
In[5]:= f[ y + f[Sqrt[y]] ]
Out[5]= 
You can do anything with f that you can do with a built-in function like Sin or Sqrt. This plots the function over the range 
to 
.
In[6]:= Plot[f[x], {x, -3, 3}]
Out[6]= 
