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Supplementary Problems in .NET
Supplementary Problems Recognize QR In .NET Framework Using Barcode Control SDK for VS .NET Control to generate, create, read, scan barcode image in Visual Studio .NET applications. Painting QR Code JIS X 0510 In .NET Using Barcode generator for VS .NET Control to generate, create QR Code image in VS .NET applications. 6.6 Determine whether the graphs of the following equations are symmetric with respect to the xaxis, the yaxis, the origin, or none of these: y2 x2 (a) y = x 2 =1 (b) y = x 3 (c) 9 4 2 + y2 = 5 2 + y2 = 9 (d) x (e) (x 1) (f ) y = (x 1)2 x+1 (i) y = (x 2 + 1)2 4 (g) 3x 2 xy + y2 = 4 (h) y = x ( j) y = x 4 3x 2 + 5 (k) y = x 5 + 7x (l) y = (x 2)3 + 1 6.7 Find an equation of the new curve when: (a) The graph of x 2 xy + y2 = 1 is re ected in the xaxis. (b) The graph of y3 xy2 + x 3 = 8 is re ected in the yaxis. (c) The graph of x 2 12x + 3y = 1 is re ected in the origin. (d) The line y = 3x + 1 is re ected in the yaxis. (e) The graph of (x 1)2 + y2 = 1 is re ected in the xaxis. Recognizing QRCode In .NET Using Barcode reader for Visual Studio .NET Control to read, scan read, scan image in .NET applications. Bar Code Creator In .NET Framework Using Barcode generation for .NET Control to generate, create bar code image in .NET framework applications. Functions and Their Graphs
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EAN13 Encoder In None Using Barcode maker for Microsoft Word Control to generate, create EAN13 image in Office Word applications. Barcode Generator In Visual C# Using Barcode generator for Visual Studio .NET Control to generate, create bar code image in VS .NET applications. FUNCTIONS AND THEIR GRAPHS
[CHAP. 7
Fig. 72 (b) The graph of the function f such that f (x) = x 2 for all x consists of all points (x, y) such that y = x 2 . This is the parabola of Fig. 32. (c) Consider the function f such that f (x) = x 3 for all x. In Fig. 73(a), we have indicated a few points on the graph, which is sketched in Fig. 73(b). Fig. 73 The numbers x for which a function f produces a value f (x) form a collection of numbers, called the domain of f . For example, the domain of the squareroot function consists of all nonnegative real numbers; the function is not de ned for negative arguments. On the other hand, the domain of the doubling function consists of all real numbers. The numbers that are the values of a function form the range of the function. The domain and the range of a function f often can be determined easily by looking at the graph of f . The domain consists of all xcoordinates of points of the graph, and the range consists of all ycoordinates of points of the graph. EXAMPLES (a) The range of the squareroot function consists of all nonnegative real numbers. Indeed, for every nonnegative real number y there is some number x such that x = y; namely, the number y2 . (b) The range of the doubling function consists of all real numbers. Indeed, for every real number y there exists a real number x such that 2x = y; namely, the number y/2. (c) Consider the absolutevalue function h, de ned by h(x) = x. The domain consists of all real numbers, but the range is made up of all nonnegative real numbers. The graph is shown in Fig. 74. When x 0, y = x is equivalent to y = x, the equation of the straight line through the origin with slope 1. When x < 0, y = x is equivalent to y = x, the equation of the straight line through the origin with slope 1. The perpendicular projection of all points of the graph onto the yaxis shows that the range consists of all y such that y 0. CHAP. 7]

