134 CHAPTER 2 Functions and Their Graphs In Problems 6–8, find the following for each function: In Problems 9–14, find the domain of each function. 9. f x x x 9 2 ( ) = − 10. f x x 2 ( ) = − 11. g x x x ( ) = 12. f x x x x2 3 2 ( ) = + − 13. f x x x 1 4 2 ( ) = + − 14. g x x x 8 ( ) = + In Problems 15–17, find f g f g f g , , , + − ⋅ and f g for each pair of functions. State the domain of each of these functions. 15. f x x g x x 2 ; 3 1 ( ) ( ) = − = + 16. f x x x g x x 3 1; 3 2 ( ) ( ) = + + = 17. f x x x g x x 1 1 ; 1 ( ) ( ) = + − = (a) f 2( ) (b) f 2 ( ) − (c) f x ( ) − (d) f x( ) − (e) f x 2 ( ) − (f) f x2( ) 6. f x x x 3 1 2 ( ) = − 7. f x x 4 2 ( ) = − 8. f x x x 4 2 2 ( ) = − 18. Find the difference quotient of f x x x 2 1; 2 ( ) = − + + that is, find f x h f x h h , 0. ( ) ( ) + − ≠ 19. Consider the graph of the function f below. (a) Find the domain and the range of f. (b) List the intercepts. (c) Find f 2 . ( ) − (d) For what value of x does f x 3? ( ) = − (e) Solve f x 0. ( ) > (f) Graph y f x 3 . ( ) = − (g) Graph y f x 1 2 . ( ) = (h) Graph y f x . ( ) = − (3, 3) (22, 21) (24, 23) x y 25 5 4 24 (0, 0) 20. Use the graph of the function f shown below to find: (a) The domain and the range of f. (b) The intervals on which f is increasing, decreasing, or constant. (c) The local minimum values and local maximum values. (d) The absolute maximum and absolute minimum. (e) Whether the graph is symmetric with respect to the x-axis, the y-axis, the origin, or none of these. (f) Whether the function is even, odd, or neither. (g) The intercepts, if any. x y 26 6 4 24 (4, 3) (24,23) (2, 21) (3, 0) (22, 1) (23, 0) In Problems 21–24, determine (algebraically) whether the given function is even, odd, or neither. 21. f x x x4 3 ( ) = − 22. g x x x 4 1 2 4 ( ) = + + 23. G x x x 1 3 ( ) = − + 24. f x x x 1 2 ( ) = + In Problems 25 and 26, use a graphing utility to graph each function over the indicated interval. Approximate any local maximum values and local minimum values. Determine where the function is increasing and where it is decreasing. 25. f x x x 2 5 1 3, 3 3 ( ) ( ) = − + − 26. f x x x x 2 5 2 1 2, 3 4 3 ( ) ( ) = − + + − 27. Find the average rate of change of f x x x 8 2 ( ) = − : (a) From 1 to 2 (b) From 0 to 1 (c) From 2 to 4 In Problems 28 and 29, find the average rate of change from 2 to 3 for each function f. Be sure to simplify. 28. f x x 2 5 ( ) = − 29. f x x x 3 4 2 ( ) = − 30. If f x x x 3 4 ,2 ( ) = − find an equation of the secant line of f from 2 to 3. In Problems 31 and 32, is the graph shown the graph of a function? 31. x y 32. x y In Problems 33 and 34, graph each function. Be sure to label at least three points. 33. f x x ( ) = 34. f x x ( ) = In Problems 35–40, graph each function using the techniques of shifting, compressing or stretching, and reflections. Identify any intercepts of the graph. State the domain and, based on the graph, find the range. 35. F x x 4 ( ) = − 36. g x x 2 ( ) = − 37. h x x 1 ( ) = − 38. f x x 1 ( ) = − 39. h x x 1 2 2 ( ) ( ) = − + 40. g x x 2 2 8 3 ( ) ( ) = − + − In Problems 41 and 42: (a) Find the domain of each function. (b) Locate any intercepts. (c) Graph each function. (d) Based on the graph, find the range. 41. f x x x x x 3 if 2 1 1 if 1 { ( ) = − < ≤ + >
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