120 CHAPTER 2 Functions and Their Graphs Other orderings of the steps shown in Example 5 would also result in the graph of f. For example, try this one: Step 1 y x 1 = Reciprocal function Step 2 y x x 3 1 3 = ⋅ = Multiply by 3; vertical stretch by a factor of 3. Step 3 y x 3 2 = − Replace x by −x 2; horizontal shift to the right 2 units. Step 4 y x 3 2 1 = − + Add 1; vertical shift up 1 unit. Figure 65 x 4 (1, 1) (2, ) (4, ) ) 1 – 2 (21, 21) y 4 24 24 x y 4 4 24 24 5 – 2 (1, 22) (3, 4) 4 4 (3, 3) 24 3 – 2 (4, x y ( ) 4, (3, 1) 1 – 2 (1, 21) x y 4 4 24 24 (1, 23) 1 ––x (a) y 5 3 ––– x –2 3 ––– x –2 (d) y 5 1 1 (c) y 5 Multiply by 3; Vertical stretch Add 1; Vertical shift up 1 unit Replace x by x 2 2; Horizontal shift right 2 units 1 ––– x –2 (b) y 5 See Figure 65. The domain of y x 1 = is { } ≠ x x 0 and its range is { } ≠ y y 0 . Because we shifted right 2 units and up 1 unit to obtain f , the domain of f is { } ≠ x x 2 and its range is { } ≠ y y 1 . Now Work PROBLEM 57 CAUTION Do not attempt to do more than one transformation at a time because this will lead to an error. Other tips: • For reflections about the y-axis, just replace x by −x, nothing more. • For shifts, use parentheses around x c ( ) − or ( ) +x c because you are replacing x by ( ) −x c or ( ) +x c . j Solution Because horizontal shifts require the form x h − , begin by rewriting f x( ) as f x x x 1 2 1 2. ( ) = − + = − + + Now use the following steps: Step 1 y x = Square root function Step 2 y x 1 = + Replace x by +x 1; horizontal shift to the left 1 unit. Step 3 y x x 1 1 = − + = − Replace x by −x; reflect about the y-axis. Step 4 y x 1 2 = − + Add 2; vertical shift up 2 units. See Figure 66. Using Graphing Techniques Graph the function f x x 1 2 ( ) = − + . Find the domain and range of f . EXAMPLE 6 Figure 66 (1, 1) 25 (4, 2) (0, 0) y 5 x 5 (0, 1) (23, 2) (1, 0) x 5 25 y 5 (0, 3) (23, 4) (1, 2) x 5 25 y 5 (0, 1) 5 25 y 5 (21, 0) (3, 2) (a) y 5 x (c) y (b) y 5 x 1 1 5 2 x 1 1 5 1 2 x (d) y 5 1 2 x 1 2 Add 2; Vertical shift up 2 units Replace x by x 1 1; Horizontal shift left 1 unit Replace x by 2x ; Reflect about y-axis The domain of f is , 1 ( ] −∞ and the range is 2, [ )∞.
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