Download Algebraic Geometry Santa Cruz 1995, Part 2: Summer Research by American Mathematical Society, János Kollár, Robert PDF

By American Mathematical Society, János Kollár, Robert Lazarsfeld

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Extra resources for Algebraic Geometry Santa Cruz 1995, Part 2: Summer Research Institute on Algebraic Geometry, July 9-29, 1995, University of California, Santa Cruz

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Notice that this graph is similar to the graph of y = sin x, except that the y-values oscillate between −2 and 2 instead of −1 and 1. 57c. In this case, the graph is similar to the graph of y = sin x except that the period is π instead of 2π (so that the oscillations occur twice as fast). 2 can be generalized. For A > 0, the graph of y = A sin x oscillates between y = −A and y = A. In this case, we call A the amplitude of the sine curve. Notice that for any positive constant c, the period of y = sin cx is 2π/c.

45. Graph f (x) = sin x and find an interval on which it is one-toone. Find the inverse of the function restricted to that interval. Graph both functions. 26. (1, 12), (2, 20), (3, 36), (4, 50), b = 32 46. Graph f (x) = cos x and find an interval on which it is one-toone. Find the inverse of the function restricted to that interval. Graph both functions. In exercises 27–36, use a graph to determine whether the function is one-to-one. If it is, graph the inverse function. In exercises 47–52, discuss whether the function described has an inverse.

45 y = x3 − 5 Most often, we cannot find a formula for an inverse function and must be satisfied with simply knowing that the inverse function exists. 5 is typical of this situation. 5 Finding Values of an Inverse Function Given that f (x) = x 5 + 8x 3 + x + 1 has an inverse, find f −1 (1) and f −1 (11). 47 (on the following page), the function looks like it might be one-to-one, but how can we be certain of this? ) Until we develop some calculus, we will be unable to verify this. 3. However, in this case, we must solve the equation y = x 5 + 8x 3 + x + 1 for x.

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