SECTION 2.2 The Graph of a Function 79 RECALL The symbol ∪ stands for union. It means the set of elements that are in either of two sets. j Figure 19 y 12 6 210 x (3, 4) (0, 24) (1, 28) (2, 0) (26, 0) (22, 0) (25, 8) (27, 28) (c) What is the range of f ? (d) List the intercepts. (Recall that these are the points, if any, where the graph crosses or touches the coordinate axes.) (e) How many times does the line y 2 = intersect the graph? (f) For what values of x does f x 8? ( ) = − (g) For what values of x is f x 0? ( ) > Solution (a) Since 5, 8 ( ) − is on the graph of f , the y-coordinate is the value of f at the x-coordinate 5; − that is, f 5 8. ( ) − = In a similar way, when x 0, = then y 4, = − so f 0 4. ( ) = − When x 3, = then y 4, = so f 3 4. ( ) = (b) To determine the domain of f , notice that the points on the graph of f have x-coordinates between 7− and 3, inclusive; and for each number x between 7− and 3, there is a point x f x , ( ) ( ) on the graph. The domain of f is x x 7 3 { } − ≤ ≤ or the interval 7, 3 . [ ] − (c) The points on the graph all have y-coordinates between 8− and 8, inclusive; and for each number y, there is at least one number x in the domain. The range of f is y y 8 8 { } − ≤ ≤ or the interval 8, 8 . [ ] − (d) The intercepts are the points 0, 4 , 6, 0 , 2, 0 , ( ) ( ) ( ) − − − and 2, 0 ( ). (e) Draw the horizontal line y 2 = on the graph in Figure 19. Notice that the line intersects the graph three times. (f) Since 7, 8 ( ) − − and 1, 8 ( ) − are the only points on the graph for which y f x 8 ( ) = = − , we have f x 8 ( ) = − when x x 7 and 1. = − = (g) To determine where f x 0 ( ) > , look at Figure 19 and determine the x-values from 7− to 3 for which the y-coordinate is positive. This occurs on 6, 2 2, 3 ( ) ( ] − − ∪ . Using inequality notation, f x 0 ( ) > for x 6 2 − < <− or x 2 3 < ≤ . When the graph of a function is given, its domain may be viewed as the shadow created by the graph on the x-axis by vertical beams of light. Its range can be viewed as the shadow created by the graph on the y-axis by horizontal beams of light. Try this technique with the graph given in Figure 19. Now Work PROBLEM 11 Obtaining Information about the Graph of a Function Consider the function: f x x x 1 2 ( ) = + + (a) Find the domain of f. (b) Is the point 1, 1 2 ( ) on the graph of f ? (c) If x 2, = what is f x ? ( ) What point is on the graph of f ? (d) If f x 2, ( ) = what is x? What point is on the graph of f ? (e) What are the x-intercepts of the graph of f (if any)? What corresponding point(s) are on the graph of f ? Solution EXAMPLE 3 (a) The domain of f is x x 2 . { } ≠ − (b) When x 1, = then f 1 1 1 1 2 2 3 ( ) = + + = f x x x 1 2 ( ) = + + The point 1, 2 3 ( ) is on the graph of f; the point 1, 1 2 ( ) is not. (continued)
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