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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1 Write each set of numbers in set-builder and interval notation, if possible. 3. 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 < x 64 The set includes all real numbers greater than 31 and less than or equal to 64. In set-builder notation this set can be described as {x 31 < x 64, x }. This notation can be read as x such that x is greater than -31 and x is less than or equal to -64, with x as an element of the real numbers. In interval notation it can be described as ( 31, 64]. This notation can be read as all values between negative 31 and 64 inclusive. The bracket indicates that the 64 is included, while the parenthesis indicates that the negative 31 is 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. 12. all multiples of 8 This set includes all integers that are multiples of 8. In set-builder notation the set can be described as {x x = 8n, n }. This notation can be read as x such that x is 8 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 1
2 24. = y 6 To determine whether this equation represents y as a function of x, solve the equation for y. 18. Each x-value is not assigned to exactly one y-value. When x = 0.04 there are two corresponding y- values, y = 449 and y = 451. Therefore, the table does not represent y as a function of x y + 4x = 11 To determine whether this equation represents y as a function of x, solve the equation for y. 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 does not pass the vertical line test, so this equation is not a function. For every x-value there will be two corresponding y- values. 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. esolutions Manual - Powered by Cognero Page 2
3 Find each function value. 30. g(x) = 2x x 14 a. g(9) b. g(3x) c. g(1 + 5m) 27. To find each value, replace x in g(x) = 2x x 14. 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. b. c. 33. a. g( 2) b. g(5x) c. g(8 4b) To find each value, replace x in. esolutions Manual - Powered by Cognero Page 3
4 36. a. g( 2) b. g(3m) c. g(4m 2) To find each value, replace m in. esolutions Manual - Powered by Cognero Page 4
5 39. State the domain of each function. When the denominator of is zero, the expression is 48. Find f ( 5) and f (12) for each piecewise function. To find f ( 5), use f (x) = 4x + 3. undefined. 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, ). To find f (12), use f (x) = 3x The square root of a negative number cannot be a real number, so 6 x 2 0. If 6 x 2 0, then. If x were greater than or less than, the expression 6 x 2 would be negative, and thus would not be a real number. The domain of h(x) is [, ]. 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 5
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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