Basics in Software
CHAPTER 1 Basics QR Code JIS X 0510 Generator In None Using Barcode creation for Software Control to generate, create Quick Response Code image in Software applications. Recognize QR Code In None Using Barcode decoder for Software Control to read, scan read, scan image in Software applications. EXAMPLE 122
Encode Quick Response Code In C# Using Barcode drawer for .NET Control to generate, create QR Code image in .NET applications. Print QRCode In Visual Studio .NET Using Barcode drawer for ASP.NET Control to generate, create QR Code image in ASP.NET applications. Let S be the set of all people and T be the set of all people Let f be the rule that assigns to each person his or her brother Is f a function QRCode Creator In Visual Studio .NET Using Barcode drawer for VS .NET Control to generate, create QR Code 2d barcode image in VS .NET applications. Make QRCode In VB.NET Using Barcode creator for .NET framework Control to generate, create QR Code 2d barcode image in .NET framework applications. SOLUTION In this case f is not a function For many people have no brother (so the rule makes no sense for them) and many people have several brothers (so the rule is ambiguous for them) EXAMPLE 123 Generating Barcode In None Using Barcode drawer for Software Control to generate, create bar code image in Software applications. Encoding UPCA In None Using Barcode drawer for Software Control to generate, create UPC A image in Software applications. Let S be the set of all people and T be the set of all strings of letters not exceeding 1500 characters (including blank spaces) Let f be the rule that assigns to each person his or her legal name (Some people have rather long names; according to the Guinness Book of World Records, the longest has 1063 letters) Determine whether f : S T is a function Draw GTIN  13 In None Using Barcode printer for Software Control to generate, create EAN 13 image in Software applications. Code 128 Code Set B Generation In None Using Barcode generator for Software Control to generate, create Code 128 Code Set A image in Software applications. SOLUTION This f is a function because every person has one and only one legal name Notice that several people may have the same name (such as JackArmstrong ), but that is allowed in the de nition of function You Try It: Let f be the rule that assigns to each real number its cube root Is this a function In calculus, the set S (called the domain of the function) and the set T (called the range of the function) will usually be sets of numbers; in fact they will often consist of one or more intervals in R The rule f will usually be given by one or several formulas Many times the domain and range will not be given explicitly These ideas will be illustrated in the examples below You Try It: Consider the rule that assigns to each real number its absolute value Is this a function Why or why not If it is a function, then what are its domain and range Making UCC.EAN  128 In None Using Barcode generation for Software Control to generate, create GS1128 image in Software applications. Painting Bar Code In None Using Barcode printer for Software Control to generate, create barcode image in Software applications. EXAMPLES OF FUNCTIONS OF A REAL VARIABLE
Paint UPC Shipping Container Symbol ITF14 In None Using Barcode drawer for Software Control to generate, create DUN  14 image in Software applications. Generate Barcode In VS .NET Using Barcode drawer for ASP.NET Control to generate, create bar code image in ASP.NET applications. EXAMPLE 124 Let S = R, T = R, and let f (x) = x 2 This is mathematical shorthand for the rule assign to each x S its square Determine whether f : R R Print Bar Code In Java Using Barcode generation for Android Control to generate, create bar code image in Android applications. Generate Code 39 Extended In Visual Studio .NET Using Barcode printer for .NET framework Control to generate, create Code 3 of 9 image in VS .NET applications. is a function
Drawing GTIN  128 In VS .NET Using Barcode creation for Reporting Service Control to generate, create GTIN  128 image in Reporting Service applications. Code 128 Printer In C# Using Barcode encoder for Visual Studio .NET Control to generate, create Code128 image in .NET applications. SOLUTION We see that f is a function since it assigns to each element of S a unique element of T namely its square Painting 2D Barcode In Visual C#.NET Using Barcode encoder for .NET framework Control to generate, create Matrix Barcode image in .NET applications. Encode GTIN  12 In Java Using Barcode generation for Java Control to generate, create UPCA Supplement 5 image in Java applications. Basics
Math Note: Notice that, in the de nition of function, there is some imprecision in the de nition of T For instance, in Example 124, we could have let T = [0, ) or T = ( 6, ) with no signi cant change in the function In the example of the name function (Example 123), we could have let T be all strings of letters not exceeding 5000 characters in length Or we could have made it all strings without regard to length Likewise, in any of the examples we could make the set S smaller and the function would still make sense It is frequently convenient not to describe S and T explicitly EXAMPLE 125 Let f (x) = + 1 x 2 Determine a domain and range for f which make f a function
SOLUTION Notice that f makes sense for x [ 1, 1] (we may not take the square root of a negative number, so we cannot allow x > 1 or x < 1) If we understand f to have domain [ 1, 1] and range R, then f : [ 1, 1] R is a function Math Note: When a function is given by a formula, as in Example 125, with no statement about the domain, then the domain is understood to be the set of all x for which the formula makes sense You Try It: Let x + 4x + 3 What are the domain and range of this function g(x) = x2 EXAMPLE 126 Let f (x) =

