SECTION 2.4 Library of Functions; Piecewise-defined Functions 103 Square Function f x x2 ( ) = Figure 42 Square Function x y 4 4 –4 (2, 4) (0, 0) (–2, 4) f(x) = x2 (1, 1) (–1, 1) The domain and the range of the identity function are the set of all real numbers. Its graph is a line with slope 1 and y -intercept 0.The line consists of all points for which the x -coordinate equals the y -coordinate.The identity function is an odd function and is increasing over its domain. Note that the graph bisects quadrants I and III. See Figure 42. The domain of the square function is the set of all real numbers; its range is the set of nonnegative real numbers.The graph of the square function is a parabola whose intercept is at 0, 0 . ( ) The square function is an even function that is decreasing on the interval , 0 ( ] −∞ and increasing on the interval 0, . [ )∞ Figure 43 Cube Function x y 4 4 24 f(x) = x3 (1, 1) (0, 0) (21, 21) 24 Cube Function f x x3 ( ) = Square Root Function f x x ( ) = Cube Root Function f x x 3 ( ) = Reciprocal Function f x x 1 ( ) = See Figure 43. The domain and the range of the cube function are the set of all real numbers. The intercept of the graph is at 0, 0 . ( ) The cube function is odd and is increasing on the interval , . ( ) −∞ ∞ See Figure 44. The domain and the range of the square root function are the set of nonnegative real numbers. The intercept of the graph is at 0, 0 . ( ) The square root function is neither even nor odd and is increasing on the interval 0, . [ )∞ See Figure 45. The domain and the range of the cube root function are the set of all real numbers. The intercept of the graph is at 0, 0 . ( ) The cube root function is an odd function that is increasing on the interval , . ( ) −∞ ∞ Refer to Example 9, page 25, for a discussion of the equation y x 1 . = See Figure 46. The domain and the range of the reciprocal function are the set of all nonzero real numbers. The graph has no intercepts. The reciprocal function is decreasing on the intervals , 0 ( ) −∞ and 0, ( )∞ and is an odd function. Figure 44 Square Root Function x y 5 2 21 f(x) = x (1, 1) (0, 0) (4, 2) Figure 45 Cube Root Function (1, 1) (21, 21) (28, 22) (8, 2) (0, 0) ( , ) 1 – 8 1 – 2 ( 2 ,2 ) 1 – 8 1 – 2 x y 3 28 23 8 f(x) = x3 Figure 46 Reciprocal Function , 2 ( ) 1 – 2 ( )1 – 2 x y 2 2 (1, 1) (21, 21) 22 22 22, 2 f(x) = 1 ––x
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