Practice Test - Chapter 1
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1 Determine whether the given relation represents y as a function of x. 1. y 3 x = 5 When x = 1, y = ±. Therefore, the relation is not one-to-one and not a function. not a function 2. The graph passes the Vertical Line Test, so the relation is a function. function 3. y = For every x-value, there is only one corresponding y-value. Therefore, the relation is one-to-one and a function. Also, the graph of the relation passes the Vertical Line Test. function esolutions Manual - Powered by Cognero Page 1
2 4. PARKING The cost of parking a car downtown is $0.75 per 30 minutes for a maximum of $4.50. Parking is charged per second. a. Write a function for c(x), the cost of parking a car for x hours. b. Find c(2.5). c. What is the domain for c(x)? Explain your reasoning. a. A cost of $0.75 per 30 minutes is equal to a cost of $1.50 per hour. The charge maximizes at $4.50 so will remain unchanged when the car is parked for longer than 3 hours. c(x) = b = 3.75 c. D = [0, 3]; Sample answer: The number of hours must be greater than or equal to 0. a. c(x) = b. $3.75 c. D = [0, 3]; Sample answer: The number of hours must be greater than or equal to 0. State the domain and range of each function. 5. The graph continues for all values of x and has a minimum y-value of 3. D = (, ), R = [ 3, ) D = (, ), R = [ 3, ) esolutions Manual - Powered by Cognero Page 2
3 6. The graph continues for all values of x such that x 5 and has a minimum y-value of 0. D = (, 5], R = [0, ) D = (, 5], R = [0, ) Find the y-intercept(s) and zeros for each function. 7. f (x) = 4x 2 8x 12 12; 1, 3 8. f (x) = x 3 + 4x 2 + 3x 0; 3, 1, 0 esolutions Manual - Powered by Cognero Page 3
4 9. MULTIPLE CHOICE Which relation is symmetric about the x-axis? 10. A x 2 yx = 2 B x 3 y = 8 C y = x D y 2 = 4x In order for a relation to be symmetric with respect to the x-axis, (x, y) must correspond with (x, y). This occurs for choice D only. (x, y) (x, y) (0.25, 1) (0.25, 1) (1, 2) (1, 2) (2.25, 3) ( 2.25, 3) (4, 4) ( 4, 4) D Determine whether each function is continuous at x = 3. If discontinuous, identify the type of discontinuity as infinite, jump, or removable. f(3) = 6,, The function is continuous at x = 3. continuous 11. f (x) = not continuous; removable discontinuity at x = 3 not continuous; removable discontinuity esolutions Manual - Powered by Cognero Page 4
5 Find the average rate of change for each function on the interval [ 2, 6]. 12. f (x) = x 4 + 3x f (x) = esolutions Manual - Powered by Cognero Page 5
6 Use the graph of each function to estimate intervals to the nearest 0.5 unit on which the function is increasing or decreasing. 14. f is increasing on (, 2.5) and decreasing on (2.5, ). f is increasing on (, 2.5) and decreasing on (2.5, ). 15. f is decreasing on (, 1.5), increasing on ( 1.5, 0), decreasing on (0, 1.5), and increasing on (1.5, ). f is decreasing on (, 1.5), increasing on ( 1.5, 0), decreasing on (0, 1.5), and increasing on (1.5, ). esolutions Manual - Powered by Cognero Page 6
7 16. Which function is shown in the graph? F f (x) = x 4 3 G f (x) = x H f (x) = x J f (x) = x The parent function g(x) = x is shifted down 3 units. Therefore, F and H are possible choices. The graph is also shifted 4 units left, which corresponds to choice H. H Identify the parent function f (x) of g(x). Then sketch the graph of g(x). 17. g(x) = (x + 3) 3 f(x) = x 3 f(x) = x 3 esolutions Manual - Powered by Cognero Page 7
8 18. g(x) = x 2 4 f(x) = x 2 f(x) = x 2 esolutions Manual - Powered by Cognero Page 8
9 19. Given f (x) = x 6 and g(x) = x 2 36, find each function and its domain. The value of x cannot equal 6 or 6. In the original expression, if x = 6 or 6, a division by 0 would occur, which is undefined. for x 6 or x [g f ](x) There are no restrictions on the domain. [f g](x) = x 2 12x esolutions Manual - Powered by Cognero Page 9
10 21. TEMPERATURE In most countries, temperature is measured in degrees Celsius. The equation that relates degrees Fahrenheit with degrees Celsius is F = C a. Write C as a function of F. b. Find two functions f and g such that C = [f g](f). a. b. When making a composition, look for the operation that must be done first. This will represent the innermost function in the composition, in this case, g(x). Sample answer: (x) = x; g(x) = x 32 a. C = (F 32) b. Sample answer: f (x) = x; g(x) = x 32 esolutions Manual - Powered by Cognero Page 10
11 Determine whether f has an inverse function. If it does, find the inverse function and state any restrictions on its domain. 22. f (x) = (x 2) 3 Graph f (x). The graph passes the Horizontal Line Test. Therefore, an inverse exists. There is no domain restriction. yes; f 1 (x) = + 2 esolutions Manual - Powered by Cognero Page 11
12 23. f (x) = Graph f (x). The graph passes the Horizontal Line Test. Therefore, an inverse exists. Graph the inverse. The domain of f 1 (x) is x 1. yes; f 1 (x) = ; x 1 esolutions Manual - Powered by Cognero Page 12
13 24. f (x) = Graph f (x). The graph passes the Horizontal Line Test. Therefore, an inverse exists. Graph the inverse. The range of f (x) = [0, ), so the domain of f 1 (x) must be limited to [0, ). yes; f 1 = 4 x 2 ; x 0 esolutions Manual - Powered by Cognero Page 13
14 25. f (x) = x 2 16 Graph f (x). The graph does not pass the Horizontal Line Test. Therefore, an inverse does not exist. no esolutions Manual - Powered by Cognero Page 14
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