By A. C. Burdette (Auth.)
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Extra resources for An Introduction to Analytic Geometry and Calculus
Note that once x is chosen, there is no ambiguity in the value of y to be used. The functional relationship exists only for x belonging to the set 5 ( — 3, — 2, —1,0, 1, 2, 3) since the association between x and y is defined only for these values. For this reason, S is said to be the domain of definition or, more simply, the domain of the function. The set of associated values of y is called the range of the function. If we use the concept of ordered pairs of numbers (x, y),'\ this relationship is completely expressed by the set of pairs ( - 3 , 0 ) , ( - 2 , 2 ) , ( - l , 1),(0, - 1 ) , ( 1 , - 2 ) , (2, - 5 ) , (3, - 9 ) .
Show that the sequence 3 4 2 3 converges to the limit 1. n+1 n 56 4. FUNCTIONS AND LIMITS What we have to show is that for any ε > 0, we can find an TV such that |s„-l|
Graph x - y = 4. (a) Intercepts: If x = 0, y = ±2i. y-axis. This is consistent with (c) below, where it is seen that x = 0 is in the excluded portion of the plane. lfy = 09x= ±2. (b) Symmetry: Theorems 3-1, 3-2, 3-3 all indicate symmetry. Hence this curve is symmetric to both axes and the origin.! (c) Extent: From Example 3-8, |JC| < 2 is excluded. (d) Additional points: The results of (a), (b), (c) make it possible to sketch the required graph from very few additional points. Observe that, owing to symmetry, the calculation of one point gives three others with no further calculation.