1-1 Functions. 3. x 4 SOLUTION: 5. 8 < x < 99 SOLUTION: 7. x < 19 or x > 21 SOLUTION: 9. { 0.25, 0, 0.25, 0.50, } SOLUTION:
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1 Write each set of numbers in set-builder and interval notation, if possible. 1. x > 50 The set includes all real numbers greater than 50. In set-builder notation this set can be described as {x x > 50, x }. This notation can be read as x such that x is greater than 50, with x as an element of the real numbers. In interval notation it can be described as (50, ). This notation can be read as all values between 50 and infinity. The parentheses indicate that the 50 is not included < x < 99 The set includes all real numbers greater than 8 and less than 99. In set-builder notation this set can be described as {x 8 < x < 99, x }. This notation can be read as x such that x is greater than 8 and x is less than 99, with x as an element of the real numbers. In interval notation it can be described as (8, 99). This notation can be read as all values between 8 and 99. The parentheses indicate that the 8 and 99 are not included. 7. x < 19 or x > x 4 The set includes all real numbers less than or equal to 4. In set-builder notation this set can be described as {x x 4, x }. This notation can be read as x such that x is less than or equal to -4, with x as an element of the real numbers. In interval notation it can be described as (, 4]. This notation can be read as all values between negative infinity and negative 4 inclusive. The bracket indicates that the -4 is included. The set includes all real numbers less than 19 or greater than 21. In set-builder notation this set can be described as {x x < 19 or x > 21, x }. This notation can be read as x such that x is less than or 19 or x is greater than 21, with x as an element of the real numbers. In interval notation it can be described as (, 19) (21, ). This notation can be read as all values between negative infinity and 19 OR between 21 and infinity. The parentheses indicate that the 19 and 21 are not included. 9. { 0.25, 0, 0.25, 0.50, } The set includes multiples of 0.25, starting with 0.25 ( 1) or In set-builder notation the set can be described as { x 0.25n = x, n 1, n }. This notation can be read as x such that x is 0.25 times n for all n greater than or equal to 1, with n as an element of the integers. The set cannot be described using one or more inequalities, and therefore cannot be written in interval notation. esolutions Manual - Powered by Cognero Page 1
2 11. x 45 or x > 86 This set includes all real numbers that are less than or equal to 45 and greater than 86. In set-builder notation the set can be described as {x x 45 or x > 86, x }. This notation can be read as x such that x is less than or equal to 45 or x is greater than 86, with x as an element of the real numbers. In interval notation it can be described as (, 45] (86, ). This notation can be read as all values between negative infinity and 45 inclusive OR between 86 and infinity. The bracket indicates that the 45 is included, while the parenthesis indicates that the 86 is not included. 13. all multiples of 5 This set includes all integers that are multiples of Each x-value is assigned to exactly one y-value. Therefore, the table represents y as a function of x. 19. = y The graph of this function passes the vertical line test. Therefore, this equation represents y as a function of x, because for every x-value there is exactly one corresponding y-value. In set-builder notation the set can be described as {x x = 5n, n }. This notation can be read as x such that x is 5 times n, with n as an element of the integers. The set cannot be described using one or more inequalities, and therefore cannot be written in interval notation. Determine whether each relation represents y as a function of x. 15. The input value x is a bank account number and the output value y is the account balance. Each value of x cannot be assigned to more than one y-value because a bank account can only have one balance at a given time. Therefore, the sentence describes y as a function of x. esolutions Manual - Powered by Cognero Page 2
3 21. 3y + 4x = 11 To determine whether this equation represents y as a function of x, solve the equation for y. 23. = x To determine whether this equation represents y as a function of x, solve the equation for y. All linear functions that can be written in slopeintercept form are non-vertical lines when graphed. Thus, they will pass the vertical line test and are functions. This equation represents y as a function of x, because for every x-value there is exactly one corresponding y-value. The graph of this equation is quadratic. All quadratic equations in which x is the squared term will open up or open down and will pass the vertical line test. All quadratic equations in which y is the squared term will open left or open right and will not pass the vertical line test. This graph passes the vertical line test, so this equation represents y as a function of x, because for every x-value there is exactly one corresponding y-value. 25. The graph passes the vertical line test. Therefore, the graph represents y as a function of x. esolutions Manual - Powered by Cognero Page 3
4 Find each function value. 31. h(y) = 3y 3 6y + 9 a. h(4) b. h( 2y) c. h(5b + 3) 27. To find each value, replace y in h(y) = 3y 3 6y + 9. a. A vertical line at x = 2 intersects the graph at more than one point. Therefore, the graph does not represent y as a function of x. 29. METEOROLOGY The five-day forecast for a city is shown. b. c. a. Represent the relation between the day of the week and the estimated high temperature as a set of ordered pairs. b. Is the estimated high temperature a function of the day of the week? the low temperature? Explain your reasoning. a. Let set A represent the days of the week and set B represent the high temperatures. Therefore, set A = {1, 2, 3, 4, 5} and set B = {70, 75, 70, 62, 65}, and the set of ordered pairs that represents the relation from set A to set B is {(1, 70), (2, 75), (3, 70), (4, 62), (5, 65)}. b. The estimated high temperature is a function of the day of the week because there is exactly one estimated high temperature each day. The estimated low temperature is also function of the day of the week because there is exactly one estimated low temperature each day. When there is exactly one output for every input, the relation is a function. 33. a. g( 2) b. g(5x) c. g(8 4b) To find each value, replace x in. esolutions Manual - Powered by Cognero Page 4
5 35. f (x) = 7 + a. f (5) b. f ( 8x) c. f (6y + 4) To find each value, replace x in f (x) = 7 +. esolutions Manual - Powered by Cognero Page 5
6 37. a. t( 4) b. t(2x) c. t(7 + n) To find each value, replace x in. 39. State the domain of each function. When the denominator of undefined. is zero, the expression is Therefore, the domain of this function is all real numbers except x = 4 and x = 1, which can be written as (, 4) ( 4, 1) ( 1, ). 41. There is no value of a that will make the expression undefined. Any value that is squared will be nonnegative. Any nonnegative number plus one will also always be nonnegative. Therefore, the domain of g(a) includes all real numbers or (, ). 43. f (a) = This function is defined only when 4a 1 > 0. The domain of f (a) is (0.25, ). 45. f (x) = + This function is defined only when x 0 and x Therefore, the function is defined for all real numbers except x = 0 and x = 1. The domain of f (x) is (, 1) ( 1, 0) (0, ). esolutions Manual - Powered by Cognero Page 6
7 47. PHYSICS The period T of a pendulum is the time for one cycle and can be calculated using the formula T = 2π, where is the length of the 51. pendulum and 9.8 is the acceleration due to gravity in meters per second squared. Is this formula a function of? If so, determine the domain. If not, explain why not. To find f ( 5), use f (x) =. To find f (12), use f (x) =. Yes; sample answer: The formula T = 2π is a function of because the length of the pendulum must be positive. With this restriction, every value of is now assigned to exactly one value of T, and the domain of the function is (0, ). Find f ( 5) and f (12) for each piecewise function. 49. To find f ( 5), use f (x) = x 2 + x + 1. To find f (12), use f (x) = x 2 + x + 1. esolutions Manual - Powered by Cognero Page 7
8 67. f (x) = x 2 6x + 8 To find f (a), replace x with a in f (x) = x 2 6x f (x) = x 5 To find f (a), replace x with a in f (x) = x 5. To find f (a + h), replace x with the expression a + h in f (x) = x 2 6x + 8. To find f (a + h), replace x with the expression a + h in f (x) = x 5. Use the expressions that you found for f (a) and f (a + h) to find. Use the expressions that you found for f (a) and f (a + h) to find. esolutions Manual - Powered by Cognero Page 10
9 71. f (x) = 7x 3 To find f (a), replace x with a in f (x) = 7x 3. envelopes have an aspect ratio (length divided by height) of 1.3 to 2.5, inclusive. The minimum allowable length is 5 inches and the maximum allowable length is 11 inches. To find f (a + h), replace x with the expression a + h in f (x) = 7x 3. Use the expressions that you found for f (a) and f (a + h) to find. a. Write the area of the envelope A as a function of length if the aspect ratio is 1.8. State the domain of the function. b. Write the area of the envelope A as a function of height h if the aspect ratio is 2.1. State the domain of the function. c. Find the area of an envelope with the maximum height at the maximum aspect ratio. a. The aspect ratio of the envelope is equal to f (x) = x 3 To find f (a), replace x with a in f (x) = x 3. The area of the envelope is A = l h. Substitute into the area equation for h. To find f (a + h), replace x with the expression a + h in f (x) = x 3. Use the expressions that you found for f (a) and f (a + h) to find. The domain of the function includes all real numbers that are greater than or equal to the minimum allowable length of 5 inches and less than or equal to the maximum allowable length of 11.5 inches. In interval notation, the domain is [5, 11.5]. b. The aspect ratio of the envelope is equal to 2.1. The area of the envelope is A = h. Substitute 2.1h into the area equation for. 75. MAIL The U.S. Postal Service requires that esolutions Manual - Powered by Cognero Page 11
10 maximum height at the maximum aspect ratio is 52.9 in 2. Use the equation for the aspect ratio to find the minimum and maximum allowable heights for the envelope. minimum height: maximum height: Determine whether each equation is a function of x. Explain. 77. x = y If x = 4, y = 4 or 4. If x = 6, y = 6 or 6. Each range value is paired with two possible domain values because it is necessary to take both the positive and negative values of the absolute value of x when solving the equation for y. Therefore, the equation does not represent a function. The domain of the function includes all real numbers that are greater than or equal to the minimum allowable height of 2.4 inches and less than or equal to the maximum allowable height of 5.5 inches. In interval notation, the domain is [2.4, 5.5]. c. It is given that the maximum aspect ratio is 2.5. It is also given that the maximum allowable length is 11.5 inches. Find the value of h when the aspect ratio is 2.5 and the length is MULTIPLE REPRESENTATIONS In this problem, you will investigate the range of a function. a. GRAPHICAL Use a graphing calculator to graph f (x) = x n for whole-number values of n from 1 to 6, inclusive. b. TABULAR Predict the range of each function based on the graph, and tabulate each value of n and the corresponding range. c. VERBAL Make a conjecture about the range of f (x) when n is even. d. VERBAL Make a conjecture about the range of f (x) when n is odd. Substitute the values that you found for l and h into the area formula. a. Graph f(x) = x n for n = 1, 2, 3, 4, 5, and 6. First graph f(x) = x. Second graph f(x) = x 2. Third graph f(x) = x 3. Fourth graph f(x) = x 4. Fifth graph f(x) = x 5. Sixth graph f(x) = x 6. Therefore, the area of an envelope with the esolutions Manual - Powered by Cognero Page 12
1-1 Functions < x 64 SOLUTION: 9. { 0.25, 0, 0.25, 0.50, } SOLUTION: 12. all multiples of 8 SOLUTION: SOLUTION:
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