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12 Code39 Reader In .NET Framework Using Barcode Control SDK for Visual Studio .NET Control to generate, create, read, scan barcode image in Visual Studio .NET applications. Creating Code 3 Of 9 In VS .NET Using Barcode creator for .NET Control to generate, create Code 3 of 9 image in Visual Studio .NET applications. 1 We want to find the inverse of the relation f (x) = 2x + 4 If we call the dependent variable y, then y = 2x + 4 Swapping the names of the variables, we get x = 2y + 4 Code 39 Scanner In .NET Framework Using Barcode reader for VS .NET Control to read, scan read, scan image in Visual Studio .NET applications. Printing Barcode In .NET Using Barcode printer for VS .NET Control to generate, create barcode image in VS .NET applications. 12
Scan Barcode In .NET Framework Using Barcode reader for Visual Studio .NET Control to read, scan read, scan image in .NET applications. Draw Code 39 In Visual C#.NET Using Barcode creator for .NET Control to generate, create Code39 image in .NET framework applications. which can be manipulated with algebra to obtain y = x/2 2 If we replace the new variable y by the relation notation f 1 (x), we get f 1 (x) = x / 2 2 2 We want to find the inverse of the relation g (x) = x 2 4x + 4 Calling the dependent variable y, we can write this as y = x 2 4x + 4 which factors to y = (x 2)2 When we swap the names of the variables, we get x = (y 2)2 Taking the complete square root of both sides produces x1/2 = y 2 which can be manipulated with algebra to obtain y = 2 x1/2 Replacing y by g 1 (x), we get g 1(x) = 2 x1/2 3 We want to find the inverse of the relation h (x) = x 3 5 Calling the dependent variable y, we have y = x3 5 Swapping the names of the variables yields x = y3 5 Encode ANSI/AIM Code 39 In .NET Framework Using Barcode printer for ASP.NET Control to generate, create Code 39 Full ASCII image in ASP.NET applications. Code 39 Full ASCII Creation In Visual Basic .NET Using Barcode creation for .NET Control to generate, create Code 39 image in .NET framework applications. 522 WorkedOut Solutions to Exercises: 1119 Make ANSI/AIM Code 39 In .NET Using Barcode drawer for .NET framework Control to generate, create Code 39 image in .NET applications. Generating EAN / UCC  13 In .NET Framework Using Barcode printer for .NET framework Control to generate, create EAN13 image in .NET framework applications. which can be morphed algebraically into y = (x + 5)1/3 Replacing y by h 1 (x), we get h 1 (x) = (x + 5)1/3 4 Let s examine each situation by imagining what happens when we try to input real numbers to the original relations, and working from there: Problem 1 and its solution: The relation and its inverse are f (x) = 2x + 4 and f 1(x) = x / 2 2 We can input any real number into either of these relations, and we always get a meaningful result Therefore, the domains of f and f 1 both span the entire set of real numbers The range of f 1 is the same as the domain of f, and the range of f is the same as the domain of f 1 Therefore, the ranges of f and f 1 both include all real numbers Problem 2 and its solution: The relation and its inverse are g (x) = x 2 4x + 4 and g 1(x) = 2 x1/2 We can input any real number we choose into g, and we always get a meaningful result However, that result is always nonnegative, because it s the square of the realnumber quantity (x 2) It s possible for the output of g to equal 0; that happens when the input is 2 The range of g is therefore the set of all nonnegative reals The domain of g 1 is identical to the range of g, so the domain of g 1 spans the set of all nonnegative reals The range of g 1 is the same as the domain of g, which is the set of all reals Problem 3 and its solution: The relation and its inverse are h (x) = x 3 5 and h 1(x) = (x + 5)1/3 We can input any real number into h, and we always get a realnumber output value Therefore, the domain of h spans the entire set of reals The same is true of the domain Drawing Barcode In .NET Using Barcode creator for Visual Studio .NET Control to generate, create barcode image in .NET applications. Code 93 Extended Creator In .NET Using Barcode creation for .NET Control to generate, create Code 9/3 image in Visual Studio .NET applications. 12
Scan Bar Code In Java Using Barcode scanner for Java Control to read, scan read, scan image in Java applications. Reading ANSI/AIM Code 39 In Visual Basic .NET Using Barcode scanner for VS .NET Control to read, scan read, scan image in .NET framework applications. of h 1; the relation is defined for all possible realnumber input values The range of h 1 is the same as the domain of h, and the range of h is the same as the domain of h 1 Therefore, the ranges of h and h 1 both span the set of all reals 5 The equation of our relation, as given in the problem, is x 2 /4 y 2 /9 = 1 Let s use algebra to morph this equation so it s stated with y all by itself on the lefthand side and an expression containing only the variable x on the righthand side If we multiply through by 36, we get 9x 2 4y 2 = 36 which can be rewritten as 4y 2 = 36 9x 2 Multiplying through by 1, we get 4y 2 = 9x 2 36 When we divide through by 4, we obtain y 2 = 9x 2 /4 9 Taking the complete square root of both sides yields y = (9x 2 /4 9)1/2 Using relation notation to express this equation and name the relation f, we get f (x) = (9x 2 /4 9)1/2 That s the relation! Now let s find its inverse We begin by restating the relation with y on the left instead of f (x), so we have y = (9x 2 /4 9)1/2 Swapping the names of the variables gives us x = (9y 2 /4 9)1/2 When we square both sides, we obtain x 2 = 9y 2 /4 9 Encode Code39 In None Using Barcode maker for Microsoft Word Control to generate, create USS Code 39 image in Word applications. UCC  12 Creation In Java Using Barcode creation for Eclipse BIRT Control to generate, create UCC128 image in BIRT reports applications. 524 WorkedOut Solutions to Exercises: 1119 Draw Bar Code In None Using Barcode maker for Excel Control to generate, create bar code image in Office Excel applications. Decode UPC  13 In Visual C# Using Barcode decoder for Visual Studio .NET Control to read, scan read, scan image in .NET framework applications. Adding 9 to each side, we get x 2 + 9 = 9y 2 /4 Multiplying through by 4, and then transposing the left and righthand sides of the equation, we come up with 9y 2 = 4x 2 + 36 Dividing through by 9 yields y 2 = 4x 2 /9 + 4 After we take the complete square root of both sides, we have y = (4x 2 /9 + 4)1/2 Replacing the variable y by the relation notation f 1 (x), we conclude that f 1 (x) = (4x 2 /9 + 4)1/2 6 With the information stated in Problem 5, along with the graph of the relation shown in Fig 122, we can see that the realnumber domain of f is the set of all reals greater than or equal to 2, or smaller than or equal to 2 Another way of stating this is to say that the domain of f is the set of all reals except those in the open interval ( 2,2) The range of f is clearly the set of all reals 7 Figure B3 shows the graph of the original relation f (x) = (9x 2 /4 9)1/2 as a pair of gray curves, and the graph of the inverse relation f 1 (x) = (4x 2 /9 + 4)1/2 as a pair of black curves The point reflector is the dashed line The realnumber domain of f 1 is the same as the realnumber range of f ; that s the set of all reals The realnumber range of f 1 is identical to the realnumber domain of f ; that s the set of all reals except those in the open interval ( 2,2) 8 Figure B4 shows the graph of the original relation with its domain restricted to the reals greater than or equal to 2 ( gray curve) and its inverse (black curve) The curve for the inverse relation is a halfhyperbola that opens upward and intersects the y axis at (0,2) A movable vertical line never intersects the black curve at more than one point Therefore, the inverse relation f 1 (x) = (4x 2 /9 + 4)1/2 is a true function of x Painting EAN 128 In VS .NET Using Barcode printer for Reporting Service Control to generate, create UCC.EAN  128 image in Reporting Service applications. Scanning UPC Symbol In None Using Barcode recognizer for Software Control to read, scan read, scan image in Software applications. 
