Chapter Test 135 The Chapter Test Prep Videos include step-by-step solutions to all chapter test exercises. These videos are available in MyLab™ Math. 1. Find the domain and range of each relation. Then determine whether each relation represents a function. (a) 2, 5 , 4, 6 , 6, 7 , 8, 8 {( )( )( )( )} (b) 1,3, 4, 2, 3,5, 1,7 ( ) ( ) ( ) ( ) { } − − (c) y 6 4 2 22 24 x 2 4 24 22 (d) y 6 4 2 22 x 2 4 24 22 In Problems 2–4, find the domain of each function and evaluate each function at x 1. = − 2. f x x 4 5 ( ) = − 3. g x x x 2 2 ( ) = + + 4. h x x x x 4 5 36 2 ( ) = − + − 5. Consider the graph of the function f below. y 4 22 24 x 4 24 (0, 2) (1, 3) (2, 0) (25, 23) (22, 0) (3, 23) (5, 22) (a) Find the domain and the range of f. (b) List the intercepts. (c) Find f 1 . ( ) (d) For what value(s) of x does f x 3? ( ) = − (e) Solve f x 0. ( ) < 6. Graph the function f x x x x 2 4 2 4 3 2 ( ) = − + + − on the interval 5, 5 ( ) − using a graphing utility. Then approximate any local maximum values and local minimum values rounded to two decimal places. Determine where the function is increasing and where it is decreasing. 7. Consider the function g x x x x x 2 1 if 1 4 if 1 ( ) = + <− − ≥− ⎧ ⎨ ⎪⎪ ⎩⎪⎪ (a) Graph the function. (b) List the intercepts. (c) Find g 5 . ( ) − (d) Find g 2 . ( ) 8. For the function f x x x 3 3 4, 2 ( ) = − + (a) Find the average rate of change of f from 3 to 4. (b) Find an equation of the secant line from 3 to 4. 9. For the functions f x x2 1 2 ( ) = + and g x x3 2, ( ) = − find the following and simplify. (a) f g x ( )( ) − (b) f g x ( )( ) ⋅ (c) f x h f x ( ) ( ) + − 10. Graph each function using the techniques of shifting, compressing or stretching, and reflecting. Start with the graph of the basic function and show all the steps. (a) h x x 2 1 3 3 ( ) ( ) = − + + (b) g x x 4 2 ( ) = + + 11. Find the difference quotient of f x x x3 . 2 ( ) = − 12. A community skating rink is in the shape of a rectangle with semicircles attached at the ends. The length of the rectangle is 20 feet less than twice the width.The thickness of the ice is 2 inches. (a) Build a model that expresses the ice volume V as a function of the width x. (b) How much ice is in the rink if the width is 90 feet? 13. Determine if the function f x x 7 2 ( ) = − − is even, odd, or neither. Chapter Test 42. f x x x x x x if 4 0 1 if 0 3 if 0 ( ) = − ≤ < = > ⎧ ⎨ ⎪⎪ ⎪ ⎩ ⎪⎪ ⎪ 43. A function f is defined by f x Ax x 5 6 2 ( ) = + − If f 1 4, ( ) = find A. 44. Constructing a Closed Box A closed box with a square base is required to have a volume of 10 cubic feet. (a) Build a model that expresses the amount A of material used to make such a box as a function of the length x of a side of the square base. (b) How much material is required for a base 1 foot by 1 foot? (c) How much material is required for a base 2 feet by 2 feet? (d) Graph A A x . ( ) = For what value of x is A smallest? 45. Area of a Rectangle A rectangle has one vertex in quadrant I on the graph of y x 10 ,2 = − another at the origin, one on the positive x-axis, and one on the positive y-axis. (a) Express the area A of the rectangle as a function of x. (b) Find the largest area A that can be enclosed by the rectangle.
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