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I had been visiting my mother for a few weeks while I did all the research So, I bought from a store near her, and had the equipment shipped to my home That sounds expensive, but, actually, the goods would have been shipped from a central warehouse no matter where I bought it And, by buying out of state, I eliminated sales tax It is a good thing we lowered the price a bit, because, when it came to cables, there was something of a shock These new high-resolution video and ber-optic audio cables cost a lot more than conventional cables And we had to buy longer cables because, as we remembered at the last minute, we had to roll the television away from the stereo cabinet when we wanted to watch it Longer cables cost a lot more than shorter ones.

Sometimes we can simply turn f inside-out to get an inverse relation, and the inverse will be a true function for all the values in the domain and range of f But often, when we seek the inverse of a function f, we get a relation that s not a true function, because some elements in the range of f map from more than one element in the domain of f When this happens, we must restrict f to define an inverse f 1 that s a true function We can usually (but not always) find a way to force f 1 to behave as a true function by excluding all values of either variable that map to more than one value of the other variable Once we ve done that, we get a bijection, ensuring that there is no ambiguity or redundancy either way

tered. They are available with a pneumatic operator to adjust the stroke automatically from a set station or a primary controller. Other positive-displacement pumps include those which move liquid with rotating gears or vanes. Their output is continuous, although noisy. But as the name implies, posit ive-displacement pumps must he allowed to discharge their rat ed flow. They must always be protected by a relief valve connected from discharge to suct,ion; otherwise, if the discharge line is restricted, high enough pressure can be developed to rupture the line or overload the motor. Cont rol of flow from a gear or vane pump may be achieved by manipulating a bypass or by regulating the discharge pressure with a bypass, both of which are shown in Fig. 9.12.

Making a relation behave as a function A little while ago, we looked at a relation whose graph is a circle with a radius of 5 units (Fig 12-5) The equation of that relation, once again, is

x 2 + y 2 = 25 which can be rewritten as y 2 = 25 x 2 and then morphed to y = (25 x 2)1/2 If we use relation notation to express this equation and name the relation f, we have f (x) = (25 x 2)1/2 The vertical-line test tells us that f is not a true function of x We can modify it so that it becomes a function of x if we restrict the range to nonnegative values Graphically, that eliminates the lower half of the circle, so that for every input value in the domain, we get only one output in the range Figure 12-8A is an illustration of this function, which we can call f+ and define as f+(x) = (25 x 2)1/2 Once again, we mustn t forget that when we take the 1/2 power of a quantity without including any sign, we mean, by default, the nonnegative square root of that quantity The solid dots indicate that the plotted points are part of the range of f+ Now suppose that we eliminate the top half of the circle including the points ( 5,0) and (5,0), getting the graph shown in Fig 12-8B The vertical-line test indicates that this is a true function of x If we call this function f , we can write f (x) = (25 x2)1/2 The white dots (small open circles) tell us that the plotted points are not part of the range of f We can restrict the range further, say to values strictly larger than 1 or values smaller than or

But, with the ups and downs of discounts and cables, the price came in just about where we had planned I made the purchases, and I was the proud owner of the pieces of a half new, half old home theater system There was one more step in the development process: scheduling delivery We gave very precise instructions on when we wanted delivery, because we would be in and out of town around the holidays Well, the message did not cross state lines, and the shipper arrived on the one day we were not there A frustrating half-hour and a few telephones straightened that out, and the system was due to be redelivered..

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