By Choquet-Bruhat Y., DeWitt-Morette C., Dillard-Bleick M.
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Extra resources for Analysis, Manifolds and Physics. Basics
5-18) is the straight line numbers. for and y = x + 3. The range is the set of all real Fig. 19 The domain is the set of all real numbers. The graph (Fig. 5-19) is the reflection in the line graph of y = x*. ] The range is the set of all real numbers. y =x of the Fig. 20 Is Fig. 5-20 the graph of a function? Since (0,0) and (0,2) are on the graph, this cannot be the graph of a function. Fig. 5-21 Fig. 21 Is Fig. 5-21 the graph of a function? Since some vertical lines cut the graph in more than one point, this cannot be the graph of a function.
F(x) = x3. Let y = x3. Then x = \/y. So, f~\y)=^/J. 72 Then y(\ - x) = 1 + x, y-yx = l + x, y - 1 = x(\ + y), x = (y -l)/(y + 1). I Let l Thus, r (y) = (y-l)/(y + l). 74 Let y = l/x. Then x = \ly. So, f~\y) = l/y. 75 Does a self-inverse function exist? Is there more than one? 74. 82, find all real roots of the given polynomial. 76 x4 - 10x2 + 9. ) = (*- 3)(* + 3)(x - l)(x + 1). Hence, the roots are 3, -3,1, -1. 77 x3 + 2x2 - 16x - 32. Inspection of the divisors of the constant term 32 reveals that -2 is a root.
Vs2 and the right half of the line y = 4 for x>2. The range consists of all real numbers < 2, plus the number 4. 16 for The domain is the set of all nonzero real numbers. The graph (Fig. 5-16) is the right half of the line x>0, plus the left half of the line y = -\ for x