Mathematica Tutor

Part 7: Functions

1. Next we define a function to assign the value 10 sin x to each x. Enter
   x=.
   f[x_]:=10*Sin[x]
   and then
   f[1]
   (Note the "_" (underscore) character which appears only on the left of the second command, and the extra colon.) You may be surprised to see
   

   10 Sin[1]

   rather than a decimal approximation. To obtain a decimal approximation, enter
   N[10*Sin[1]]
   The command "N" stands for "numerical value."

You should notice two important things here:

• Functions that are built into Mathematica always start with capital letters. Functions that you define can start with either capital or lower-case letters.
  
  
• The arguments of functions in Mathematica are always enclosed in square brackets, not parentheses as you might expect. Parentheses in Mathematica are used only to indicate the grouping of terms.
  

2. To be sure that the action of N is clear, enter
   103751/2053
   and then
   N[103751/2053]
   
   
3. An alternate way of forcing Mathematica to return a decimal approximation to f[1] is to enter
   f[1.0]
   
   
4. Note that Mathematica distinguishes between a function and an expression. Enter
   g[x]=10*Sin[x]
   Then enter
   g[1]
   followed by
   g[x]
   The symbols g[x] stand for the expression, not the function. Try the same thing with g[x]:=10*Sin[x]
   (Colon, but no underscore.) You should get almost the same results. Leaving out the underscore when you define a function is a common source of problems when using Mathematica.
   
   
5. Let's evaluate f at pi/6. First, we need the value of pi. Enter
   Pi
   Alter this line to read
   N[Pi]
   

   Now find a decimal approximation to f(pi/6).
   

6. Find a decimal approximation to each of the following: f(2) and f(pi/3).