SECTION 2.4 Library of Functions; Piecewise-defined Functions 101 (continued) From the results of Example 1 and Figure 38, we have the following properties of the cube root function. Properties of the Cube Root Function ( ) = f x x 3 • The domain and the range of f x x 3 ( ) = are the set of all real numbers. • The x -intercept of the graph of f x x 3 ( ) = is 0. • The y -intercept of the graph of f x x 3 ( ) = is also 0. • f x x 3 ( ) = is odd. The graph is symmetric with respect to the origin. • f x x 3 ( ) = is increasing on the interval , . ( ) −∞ ∞ • f x x 3 ( ) = does not have any local minima or any local maxima. Graphing the Cube Root Function (a) Determine whether f x x 3 ( ) = is even, odd, or neither. State whether the graph of f is symmetric with respect to the y -axis, symmetric with respect to the origin, or neither. (b) Find the intercepts, if any, of the graph of f x x. 3 ( ) = (c) Graph f x x. 3 ( ) = Solution EXAMPLE 1 (a) Because f x x x f x 3 3 ( ) ( ) − = − = − = − the cube root function is odd. The graph of f is symmetric with respect to the origin. (b) The y -intercept is f 0 0 0. 3 ( ) = = The x -intercept is found by solving the equation f x 0. ( ) = f x x x 0 0 0 3 ( ) = = = The x -intercept is also 0. (c) Use the function to form Table 5 and obtain some points on the graph. Because of the symmetry with respect to the origin, we find only points x y , ( ) for which x 0. ≥ Figure 38 shows the graph of f x x. 3 ( ) = f x x 3 ( ) = Cube both sides of the equation. x ( ) = = 3 y f x x x, y ( ) 0 0 ( ) 0, 0 1 8 1 2 ( ) 1 8 , 1 2 1 1 ( ) 1, 1 8 2 ( ) 8, 2 Table 5 Figure 38 Cube Root Function (1, 1) (21, 21) (8, 2) (0, 0) ( , ) 1 – 8 1 – 2 ( 2 ,2 ) 1 – 8 1 – 2 x y 3 28 23 8 (28, 22) Graphing the Absolute Value Function (a) Determine whether f x x ( ) = is even, odd, or neither. State whether the graph of f is symmetric with respect to the y -axis, symmetric with respect to the origin, or neither. (b) Determine the intercepts, if any, of the graph of f x x . ( ) = (c) Graph f x x . ( ) = EXAMPLE 2
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