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A function is a correspondence between two sets of numbers A and B such that for each number in A there corresponds a unique number in B. For example, the squaring function: For each real number there corresponds a unique non-negative real number called its square. While it is customary to write f(x) = x2, one must understand that there is no special significance to the letter x. It is the process of squaring that defines the function. Although Mathematica allows a function to be defined in terms of a variable, as in f[x_]= x2, the variable x acts as a dummy and is insignificant. The function would be the same had we used y, z, or any other symbol. A pure function is defined without reference to any specific variable. Its arguments are labeled #1, #2, #3, and so forth. To distinguish a pure function from any other Mathematica construct, an ampersand, &, is used at the end of its definition. Once defined, we can deal with a pure function as we would any other function. Although the concept of a pure function is a natural one, it is possible to use Mathematica and never be concerned with it. Occasionally, however, Mathematica will express an answer as a pure function and it is therefore worthy of a brief mention. The interested reader can find more information in Mathematica s Documentation Center.

iTunes uses icons similar to other programs on the iPad, so getting around is quite easy. There are three buttons at the top and seven icons or soft keys at the bottom to help you. Look at Figure 19 1 to see the soft keys and features. Scrolling is just like scrolling in any other program; move your finger up or down to look at the selections available.

1 heads _

f = #12 &; f[3] 9 f[x] x2 f[a + b] (a + b)2

Function[x, body]is a pure function with a single parameter x. Function[{x1, x2,...}, body] is a pure function with a list of parameters x1, x2,...

Along the top of the iTunes music store screen are three buttons: Featured, Top Charts, and Genius. By default, you are shown the Featured selections when you start iTunes.

d2y + y = 0; y '(0) = y(0) = 1 dx 2 as a pure function and evaluate it for x = /4. DSolve[{y''[x]+ y[x] 0, y[0] 1, y [0]= 1}, y, x]

Function[{x}, Cos[x] + Sin[x]]}} Function[{x}, Cos[x]+ Sin[x]][o/4]

1 heads _

If you like to see what is popular in a particular category, you will want to browse the Top Charts category. Tap Top Charts at the top, and then tap a category or genre to see what is popular for that category. CAUTION: These songs or videos are selling well, but that doesn t mean that they will appeal to you. Always give the item a preview and check out the reviews before you pay for it.

A.1 Express as a pure function the process of adding the square of a number to its square root and compute its value at 9.

f = #1^2 + Sqrt[#1] &; f[9] 84

Tap the Genre button in the upper left corner to select a particular genre. For example, if you touch Singer Songwriter, you will see the top ten songs or albums in that category only. The initial view shows you the Top Songs on the left and Top Albums on the right. You simply scroll as you would in any other program on the iPad.

A.2 A number is formed from two other numbers by adding the square of their sum to the sum of their squares. Express this operation as a pure function and compute its value for the numbers 3 and 4.

Fig. 6-1

g =(#1 + #2)^2 + #1^2 + #2^2 &; g[3, 4] 74

Touch the Genres button to browse music based on a genre. This is particularly helpful if you have a favorite type of music and would like to browse just that category. There is quite an extensive list of genres to browse; just scroll down the list as you would in any other iPad program.

A.3 Express the derivative of the function Sin as a pure function and compute its vale at /6.

f = Sin' Cos[#1]& f[o/6] 3 2

Go ahead and browse through the music until you see something that you would like to preview or buy.