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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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