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Supplementary Problems
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6.6 Determine whether the graphs of the following equations are symmetric with respect to the x-axis, the y-axis, 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 x-axis. (b) The graph of y3 xy2 + x 3 = 8 is re ected in the y-axis. (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 y-axis. (e) The graph of (x 1)2 + y2 = 1 is re ected in the x-axis.
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Functions and Their Graphs
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7.1 THE NOTION OF A FUNCTION To say that a quantity y is a function of some other quantity x means, in ordinary language, that the value of y depends on the value of x. For example, the volume V of a cube is a function of the length s of a side. In fact, the dependence of V on s can be made precise through the formula V = s3 . Such a speci c association of a number s3 with a given number s is what mathematicians usually mean by a function. In Fig. 7-1, we picture a function f as some sort of process which, from a number x, produces a number f (x); the number x is called an argument of f and the number f (x) is called the value of f for the argument x.
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Fig. 7-1 EXAMPLES
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(a) The square-root function associates with each nonnegative real number x the value number y such that y2 = x. x; that is, the unique nonnegative real
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(b) The doubling function g associates with each real number x the value 2x. Thus, g(3) = 6, g( 1) = 2, g( 2) = 2 2.
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The graph of a function f consists of all points (x, y) such that y = f (x). Thus, the graph of f is the graph of the equation y = f (x). EXAMPLES
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(a) Consider the function f such that f (x) = x + 1 for all x. The graph of f is the set of all points (x, y) such that y = x + 1. This is a straight line, with slope 1 and y-intercept 1 (see Fig. 7-2). 49
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FUNCTIONS AND THEIR GRAPHS
[CHAP. 7
Fig. 7-2
(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. 3-2. (c) Consider the function f such that f (x) = x 3 for all x. In Fig. 7-3(a), we have indicated a few points on the graph, which is sketched in Fig. 7-3(b).
Fig. 7-3 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 square-root 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 x-coordinates of points of the graph, and the range consists of all y-coordinates of points of the graph. EXAMPLES
(a) The range of the square-root 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 absolute-value 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. 7-4. 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 y-axis shows that the range consists of all y such that y 0.
CHAP. 7]
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