Part Two 401

USS Code 39 Recognizer In .NETUsing Barcode Control SDK for .NET Control to generate, create, read, scan barcode image in .NET applications.

Making Code 3/9 In .NET FrameworkUsing Barcode maker for Visual Studio .NET Control to generate, create Code39 image in .NET applications.

y 6 4 f 2 x 6 4 2 2 4 g 6 h 2 4 6

Scanning Code 3 Of 9 In .NET FrameworkUsing Barcode reader for VS .NET Control to read, scan read, scan image in .NET framework applications.

Creating Barcode In .NETUsing Barcode generator for VS .NET Control to generate, create barcode image in .NET applications.

Figure 20-1

Scanning Bar Code In .NETUsing Barcode reader for .NET framework Control to read, scan read, scan image in VS .NET applications.

Create Code 39 Extended In Visual C#.NETUsing Barcode generator for Visual Studio .NET Control to generate, create Code 39 Extended image in Visual Studio .NET applications.

Illustration for Question and Answer 11-7

Printing Code 3 Of 9 In VS .NETUsing Barcode generator for ASP.NET Control to generate, create ANSI/AIM Code 39 image in ASP.NET applications.

Code 3 Of 9 Printer In VB.NETUsing Barcode maker for .NET Control to generate, create Code 39 Extended image in .NET applications.

variable is represented by r (the radial distance from the origin) How can we tell if the relation is a function of q

Creating UPC Symbol In .NETUsing Barcode generation for Visual Studio .NET Control to generate, create UCC - 12 image in .NET applications.

Creating Code 128C In Visual Studio .NETUsing Barcode encoder for VS .NET Control to generate, create Code 128C image in Visual Studio .NET applications.

Answer 11-8

Code39 Drawer In Visual Studio .NETUsing Barcode maker for VS .NET Control to generate, create ANSI/AIM Code 39 image in Visual Studio .NET applications.

Drawing Monarch In Visual Studio .NETUsing Barcode generator for .NET Control to generate, create ANSI/AIM Codabar image in Visual Studio .NET applications.

We can draw the graph of the relation in a Cartesian plane, plotting values of q along the horizontal axis, and plotting values of r along the vertical axis We can allow both q and r to attain all possible real-number values Then we can use the Cartesian vertical-line test to see if the relation is a function of q

Print UCC - 12 In JavaUsing Barcode encoder for Java Control to generate, create GTIN - 12 image in Java applications.

USS-128 Generation In NoneUsing Barcode encoder for Software Control to generate, create GS1-128 image in Software applications.

Question 11-9

Code 128 Code Set B Creation In Objective-CUsing Barcode maker for iPad Control to generate, create Code 128 Code Set B image in iPad applications.

Code 3 Of 9 Reader In Visual C#.NETUsing Barcode recognizer for VS .NET Control to read, scan read, scan image in .NET framework applications.

How do functions add, subtract, multiply, and divide

UCC - 12 Printer In Objective-CUsing Barcode maker for iPad Control to generate, create UCC-128 image in iPad applications.

Matrix 2D Barcode Maker In C#Using Barcode maker for VS .NET Control to generate, create 2D Barcode image in .NET framework applications.

Answer 11-9

Code128 Reader In Visual C#.NETUsing Barcode recognizer for VS .NET Control to read, scan read, scan image in .NET framework applications.

Draw Code 128 Code Set C In NoneUsing Barcode creator for Microsoft Word Control to generate, create Code 128 Code Set A image in Word applications.

To add one function to another, we add both sides of their equations This can be done in either order, producing identical results If f1 and f2 are functions of x, then ( f1 + f2)(x) = f1(x) + f2(x) and ( f2 + f1)(x) = f2(x) + f1(x)

Review Questions and Answers

To subtract one function from another, we subtract both sides of their equations This can be done in either order, usually producing different results If f1 and f2 are functions of x, then ( f1 f2)(x) = f1(x) f2(x) and ( f2 f1)(x) = f2(x) f1(x) To multiply one function by another, we multiply both sides of their equations This can be done in either order, producing identical results If f1 and f2 are functions of x, then ( f1 f2)(x) = f1(x) f2(x) and ( f2 f1)(x) = f2(x) f1(x) To divide one function by another, we divide both sides of their equations This can be done in either order, usually producing different results If f1 and f2 are functions of x, then ( f1/f2)(x) = f1(x)/f2(x) and ( f2/f1)(x) = f2(x)/f1(x)

Question 11-10

When we add, subtract, multiply, or divide functions, we must adhere to three important rules What are they

Answer 11-10

First, we must be sure that the functions both operate on the same thing In other words, the independent variables must describe the same parameters or phenomena Second, we must restrict the domain of the resultant function to only those values that are in the domains of both functions (the intersection of the domains) Third, if we divide a function by another function, we can t define the resultant function for any value of the independent variable where the denominator function becomes 0

12

Question 12-1

How can we informally define the inverse of a relation

Part Two 403 Answer 12-1

The inverse of a relation is another relation that undoes whatever the original relation does Also, the original relation undoes whatever its inverse does

Question 12-2

How can we rigorously define the inverse of a relation

Answer 12-2

Let f be a relation where x is the independent variable and y is the dependent variable The inverse relation for f is another relation f 1 such that f 1 [ f (x)] = x for all values of x in the domain of f, and f [f for all values of y in the range of f

Question 12-3

(y)] = y

Suppose we ve drawn the graph of a relation f in the Cartesian xy plane How can we create the graph of the inverse relation f 1

Answer 12-3

Imagine the line y = x as a point reflector For any point on the graph of f, its counterpoint on the graph of f 1 lies on the opposite side of the line y = x but the same distance away, as shown in Fig 20-2 Mathematically, we can do this transformation by reversing the sequence of the ordered pair representing the point When we want to obtain the graph of f 1 based on the graph of f, we can flip the whole graph over in three dimensions around the line y = x, as if that line were the hinge of a revolving door

Question 12-4