102 CHAPTER 2 Functions and Their Graphs Solution (a) Because f x x x f x ( ) ( ) − = − = = the absolute value function is even. The graph of f is symmetric with respect to the y -axis. (b) The y -intercept is f 0 0 0. ( ) = = The x -intercept is found by solving the equation f x x 0. ( ) = = The x -intercept is 0. (c) Use the function to form Table 6 and obtain some points on the graph. Because of the symmetry with respect to the y -axis, we only need to find points x y , ( ) for which x 0. ≥ Figure 39 shows the graph of f x x . ( ) = x y f x x ( ) = = x, y ( ) 0 0 ( ) 0, 0 1 1 ( ) 1, 1 2 2 ( ) 2, 2 3 3 ( ) 3, 3 Table 6 Figure 39 Absolute Value Function x y 3 2 1 (1, 1) (21, 1) (2, 2) (22, 2) (3, 3) (23, 3) (0, 0) 21 22 23 21 3 1 2 Seeing the Concept Graph = y x on a square screen and compare what you see with Figure 39. Note that some graphing utilities use ( )x abs for absolute value. Properties of the Absolute Value Function ( ) = f x x • The domain of ( ) = f x x is the set of all real numbers. • The range of f is y y 0 . { } ≥ • The x -intercept of the graph of ( ) = f x x is 0. • The y -intercept of the graph of f x x ( ) = is also 0. • f x x ( ) = is even. The graph is symmetric with respect to the y -axis. • f x x ( ) = is decreasing on the interval , 0 . ( ] −∞ • f x x ( ) = is increasing on the interval 0, . [ )∞ • f x x ( ) = has an absolute minimum of 0 at x 0. = From Example 2 and Figure 39, we have the following properties of the absolute value function. Library of Functions Below is a list of the key functions that we have discussed. In going through this list, pay special attention to the properties of each function, particularly to the domain of each function and to the shape of each graph. Knowing these graphs, along with key points on each graph, lays the foundation for further graphing techniques. Identity Function f x x ( ) = Constant Function f x b b is a real number ( ) = Figure 40 Constant Function x y f(x) = b (0, b) Figure 41 Identity Function x y 3 3 –3 (1, 1) (–1, –1) f(x) = x (0, 0) See Figure 40. The domain of a constant function is the set of all real numbers; its range is the set consisting of a single number b. Its graph is a horizontal line whose y -intercept is b. The constant function is an even function. See Figure 41.
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