 Home
 Products
 Integration
 Tutorial
 Barcode FAQ
 Purchase
 Company
FUNCTIONS AND THEIR GRAPHS in .NET
FUNCTIONS AND THEIR GRAPHS QRCode Decoder In .NET Using Barcode Control SDK for Visual Studio .NET Control to generate, create, read, scan barcode image in VS .NET applications. Denso QR Bar Code Generation In VS .NET Using Barcode encoder for VS .NET Control to generate, create QR image in .NET applications. [CHAP. 7
QR Code JIS X 0510 Decoder In .NET Using Barcode scanner for VS .NET Control to read, scan read, scan image in .NET applications. Creating Barcode In Visual Studio .NET Using Barcode generator for .NET framework Control to generate, create bar code image in .NET applications. Applying the quadratic formula (see the algebra insert in Problem 5.2), ( 2)2 4(10) 2 36 2 36 1 = = 2 2 2 2 6 1 = 1 3 1 = 2 Thus, the other two roots of f (x) are the complex numbers 1 + 3 1 and 1 3 1. x= ( 2) (b) Let us nd the roots of f (x) = x 3 5x 2 + 3x + 9. The integral roots (if any) must be divisors of 9: 1, 3, 9. 1 is not a root, but 1 is a root. Hence f (x) is divisible by x + 1 [see Fig. 710(b)]. The other roots of f (x) must be the roots of x 2 6x + 9. But x 2 6x + 9 = (x 3)2 . Thus, 1 and 3 are the roots of f (x); 3 is called a repeated root because (x 3)2 is a factor of x 3 5x 2 + 3x + 9. Barcode Recognizer In .NET Framework Using Barcode reader for VS .NET Control to read, scan read, scan image in .NET framework applications. QRCode Creator In C#.NET Using Barcode encoder for .NET framework Control to generate, create QRCode image in VS .NET applications. Theorem 7.3: (Fundamental Theorem of Algebra): of degree n has exactly n roots.
Paint Quick Response Code In VS .NET Using Barcode encoder for ASP.NET Control to generate, create Quick Response Code image in ASP.NET applications. Painting QR Code ISO/IEC18004 In Visual Basic .NET Using Barcode drawer for Visual Studio .NET Control to generate, create Quick Response Code image in Visual Studio .NET applications. If repeated roots are counted multiply, then every polynomial
Data Matrix ECC200 Generation In VS .NET Using Barcode creation for .NET framework Control to generate, create Data Matrix ECC200 image in .NET framework applications. Print Code 128C In Visual Studio .NET Using Barcode creation for .NET Control to generate, create Code 128 image in Visual Studio .NET applications. EXAMPLE In example (b) above, the polynomial x3 5x2 + 3x + 9 of degree 3 has two roots, 1 and 3, but 3 is a repeated Creating 2D Barcode In Visual Studio .NET Using Barcode printer for .NET framework Control to generate, create Matrix Barcode image in .NET applications. UPC Shipping Container Symbol ITF14 Drawer In Visual Studio .NET Using Barcode encoder for .NET Control to generate, create UPC Case Code image in Visual Studio .NET applications. root of multiplicity 2 and, therefore, is counted twice. A proof of Theorem 7.3 cannot be given here because it requires knowledge of the theory of functions of a complex variable. Barcode Decoder In Java Using Barcode reader for Java Control to read, scan read, scan image in Java applications. DataBar Creator In Java Using Barcode drawer for Java Control to generate, create GS1 RSS image in Java applications. Since the complex roots of a polynomial with real coef cients occur in pairs, a b 1, the polynomial can have only an even number (possibly zero) of complex roots. Hence, a polynomial of odd degree must have at least one real root. Draw European Article Number 13 In Java Using Barcode printer for Java Control to generate, create EAN13 Supplement 5 image in Java applications. Code39 Generator In VS .NET Using Barcode encoder for ASP.NET Control to generate, create Code 3 of 9 image in ASP.NET applications. Solved Problems
Print Barcode In Visual Studio .NET Using Barcode creation for Reporting Service Control to generate, create barcode image in Reporting Service applications. Printing ECC200 In None Using Barcode drawer for Online Control to generate, create Data Matrix ECC200 image in Online applications. 7.1 Find the domain and the range of the function f such that f (x) = x 2 .
ANSI/AIM Code 128 Maker In Java Using Barcode generation for Java Control to generate, create ANSI/AIM Code 128 image in Java applications. Paint UPCA In Java Using Barcode encoder for Java Control to generate, create UPCA Supplement 2 image in Java applications. Since x 2 is de ned for every real number x, the domain of f consists of all real numbers. To nd the range, notice that x 2 0 for all x and, therefore, x 2 0 for all x. Every nonpositive number y appears as a value x 2 for a suitable argument x; namely, for the argument x = y (and also for the argument x = y). Thus the range of f is ( , 0]. This can be seen more easily by looking at the graph of y = x 2 [see Fig. 711(a)]. Fig. 711 7.2 Find the domain and the range of the function f de ned by f (x) = x + 1 if 1 < x < 0 x if 0 x < 1 CHAP. 7] FUNCTIONS AND THEIR GRAPHS
The domain of f consists of all x such that either 1 < x < 0 or 0 x < 1. This makes up the open interval ( 1, 1). The range of f is easily found from the graph in Fig. 711(b), whose projection onto the yaxis is the halfopen interval [0, 1). 7.3 De ne f (x) as the greatest integer less than or equal to x; this value is usually denoted by [x]. Find the domain and the range, and draw the graph of f . Since [x] is de ned for all x, the domain is the set of all real numbers. The range of f consists of all integers. Part of the graph is shown in Fig. 712. It consists of a sequence of horizontal, halfopen unit intervals. (A function whose graph consists of horizontal segments is called a step function.) Fig. 712 Fig. 713 7.4 Consider the function f de ned by the formula f (x) = x2 1 x 1
whenever this formula makes sense. Find the domain and the range, and draw the graph of f .
The formula makes sense whenever the denominator x 1 is not 0. Hence, the domain of f is the set of all real numbers different from 1. Now x 2 1 = (x 1)(x + 1) and so x2 1 =x+1 x 1 Hence, the graph of f (x) is the same as the graph of y = x + 1, except that the point corresponding to x = 1 is missing (see Fig. 713). Thus, the range consists of all real numbers except 2. 7.5 (a) Show that a set of points in the xyplane is the graph of some function of x if and only if the set intersects every vertical line in at most one point (vertical line test). (b) Determine whether the sets of points indicated in Fig. 714 are graphs of functions.

