You used set notation to denote elements subsets
You used set notation to denote elements, subsets, and complements. (Lesson 0 -1) • Describe subsets of real numbers. • Identify and evaluate functions and state their domains.
• set-builder notation • interval notation • function notation • independent variable • implied domain • piecewise-defined function • relevant domain
Use Set-Builder Notation A. Describe {2, 3, 4, 5, 6, 7} using set-builder notation. The set includes natural numbers greater than or equal to 2 and less than or equal to 7. This is read as the set of all x such that 2 is less than or equal to x and x is less than or equal to 7 and x is an element of the set of natural numbers. Answer:
Use Set-Builder Notation A. Describe {2, 3, 4, 5, 6, 7} using set-builder notation. The set includes natural numbers greater than or equal to 2 and less than or equal to 7. This is read as the set of all x such that 2 is less than or equal to x and x is less than or equal to 7 and x is an element of the set of natural numbers. Answer:
Use Set-Builder Notation B. Describe x > – 17 using set-builder notation. The set includes all real numbers greater than – 17. Answer:
Use Set-Builder Notation B. Describe x > – 17 using set-builder notation. The set includes all real numbers greater than – 17. Answer:
Use Set-Builder Notation C. Describe all multiples of seven using set-builder notation. The set includes all integers that are multiples of 7. Answer:
Use Set-Builder Notation C. Describe all multiples of seven using set-builder notation. The set includes all integers that are multiples of 7. Answer:
Describe {6, 7, 8, 9, 10, …} using set-builder notation. A. B. C. D.
Describe {6, 7, 8, 9, 10, …} using set-builder notation. A. B. C. D.
Use Interval Notation A. Write – 2 ≤ x ≤ 12 using interval notation. The set includes all real numbers greater than or equal to – 2 and less than or equal to 12. Answer:
Use Interval Notation A. Write – 2 ≤ x ≤ 12 using interval notation. The set includes all real numbers greater than or equal to – 2 and less than or equal to 12. Answer: [– 2, 12]
Use Interval Notation B. Write x > – 4 using interval notation. The set includes all real numbers greater than – 4. Answer:
Use Interval Notation B. Write x > – 4 using interval notation. The set includes all real numbers greater than – 4. Answer: (– 4, )
Use Interval Notation C. Write x < 3 or x ≥ 54 using interval notation. The set includes all real numbers less than 3 and all real numbers greater than or equal to 54. Answer:
Use Interval Notation C. Write x < 3 or x ≥ 54 using interval notation. The set includes all real numbers less than 3 and all real numbers greater than or equal to 54. Answer:
Write x > 5 or x < – 1 using interval notation. A. B. C. (– 1, 5) D.
Write x > 5 or x < – 1 using interval notation. A. B. C. (– 1, 5) D.
Identify Relations that are Functions A. Determine whether the relation represents y as a function of x. The input value x is the height of a student in inches, and the output value y is the number of books that the student owns. Answer:
Identify Relations that are Functions A. Determine whether the relation represents y as a function of x. The input value x is the height of a student in inches, and the output value y is the number of books that the student owns. Answer: No; there is more than one y-value for an x-value.
Identify Relations that are Functions B. Determine whether the table represents y as a function of x. Answer:
Identify Relations that are Functions B. Determine whether the table represents y as a function of x. Answer: No; there is more than one y-value for an x-value.
Identify Relations that are Functions C. Determine whether the graph represents y as a function of x. Answer:
Identify Relations that are Functions C. Determine whether the graph represents y as a function of x. Answer: Yes; there is exactly one y-value for each xvalue. Any vertical line will intersect the graph at only one point. Therefore, the graph represents y as a function of x.
Identify Relations that are Functions D. Determine whether x = 3 y 2 represents y as a function of x. To determine whether this equation represents y as a function of x, solve the equation for y. x = 3 y 2 Original equation Divide each side by 3. Take the square root of each side.
Identify Relations that are Functions This equation does not represent y as a function of x because there will be two corresponding y-values, one positive and one negative, for any x-value greater than 0. Let x = 12. Answer:
Identify Relations that are Functions This equation does not represent y as a function of x because there will be two corresponding y-values, one positive and one negative, for any x-value greater than 0. Let x = 12. Answer: No; there is more than one y-value for an x-value.
Determine whether 12 x 2 + 4 y = 8 represents y as a function of x. A. Yes; there is exactly one y-value for each x-value. B. No; there is more than one y-value for an x-value.
Determine whether 12 x 2 + 4 y = 8 represents y as a function of x. A. Yes; there is exactly one y-value for each x-value. B. No; there is more than one y-value for an x-value.
Find Function Values A. If f (x) = x 2 – 2 x – 8, find f (3). To find f (3), replace x with 3 in f (x) = x 2 – 2 x – 8 Original function f (3) = 3 2 – 2(3) – 8 Substitute 3 for x. =9– 6– 8 Simplify. = – 5 Subtract. Answer:
Find Function Values A. If f (x) = x 2 – 2 x – 8, find f (3). To find f (3), replace x with 3 in f (x) = x 2 – 2 x – 8 Original function f (3) = 3 2 – 2(3) – 8 Substitute 3 for x. =9– 6– 8 Simplify. = – 5 Subtract. Answer: – 5
Find Function Values B. If f (x) = x 2 – 2 x – 8, find f (– 3 d). To find f (– 3 d), replace x with – 3 d in f (x) = x 2 – 2 x – 8 f (– 3 d) = (– 3 d)2 – 2(– 3 d) – 8 = 9 d 2 + 6 d – 8 Answer: Original function Substitute – 3 d for x. Simplify.
Find Function Values B. If f (x) = x 2 – 2 x – 8, find f (– 3 d). To find f (– 3 d), replace x with – 3 d in f (x) = x 2 – 2 x – 8 f (– 3 d) = (– 3 d)2 – 2(– 3 d) – 8 = 9 d 2 + 6 d – 8 Answer: 9 d 2 + 6 d – 8 Original function Substitute – 3 d for x. Simplify.
Find Function Values C. If f (x) = x 2 – 2 x – 8, find f (2 a – 1). To find f (2 a – 1), replace x with 2 a – 1 in f (x) = x 2 – 2 x – 8 f (2 a – 1) = (2 a – 1)2 – 2(2 a – 1) – 8 Original function Substitute 2 a – 1 for x. = 4 a 2 – 4 a + 1 – 4 a + 2 – 8 Expand (2 a – 1)2 and 2(2 a – 1). = 4 a 2 – 8 a – 5 Answer: Simplify.
Find Function Values C. If f (x) = x 2 – 2 x – 8, find f (2 a – 1). To find f (2 a – 1), replace x with 2 a – 1 in f (x) = x 2 – 2 x – 8 f (2 a – 1) = (2 a – 1)2 – 2(2 a – 1) – 8 Original function Substitute 2 a – 1 for x. = 4 a 2 – 4 a + 1 – 4 a + 2 – 8 Expand (2 a – 1)2 and 2(2 a – 1). = 4 a 2 – 8 a – 5 Answer: 4 a 2 – 8 a – 5 Simplify.
If A. B. C. D. , find f (6).
If A. B. C. D. , find f (6).
Find Domains Algebraically A. State the domain of the function . Because the square root of a negative number cannot be real, 4 x – 1 ≥ 0. Therefore, the domain of g(x) is all real numbers x such that x ≥ Answer: , or .
Find Domains Algebraically A. State the domain of the function . Because the square root of a negative number cannot be real, 4 x – 1 ≥ 0. Therefore, the domain of g(x) is all real numbers x such that x ≥ , or Answer: all real numbers x such that x ≥ or . ,
Find Domains Algebraically B. State the domain of the function When the denominator of . is zero, the expression is undefined. Solving t 2 – 1 = 0, the excluded values in the domain of this function are t = 1 and t = – 1. The domain of this function is all real numbers except t = 1 and t = – 1, or Answer: .
Find Domains Algebraically B. State the domain of the function When the denominator of . is zero, the expression is undefined. Solving t 2 – 1 = 0, the excluded values in the domain of this function are t = 1 and t = – 1. The domain of this function is all real numbers except t = 1 and t = – 1, or Answer: .
Find Domains Algebraically C. State the domain of the function This function is defined only when 2 x – 3 > 0. Therefore, the domain of f (x) is or Answer: . .
Find Domains Algebraically C. State the domain of the function This function is defined only when 2 x – 3 > 0. Therefore, the domain of f (x) is or Answer: . or .
State the domain of g (x) = A. B. C. D. or [4, ∞) or [– 4, 4] or (− , − 4] .
State the domain of g (x) = A. B. C. D. or [4, ∞) or [– 4, 4] or (− , − 4] .
Evaluate a Piecewise-Defined Function A. FINANCE Realtors in a metropolitan area studied the average home price per square foot as a function of total square footage. Their evaluation yielded the following piecewisedefined function. Find the average price per square foot for a home with the square footage of 1400 square feet.
Evaluate a Piecewise-Defined Function Because 1400 is between 1000 and 2600, use to find p(1400). Function for 1000 ≤ a < 2600 Substitute 1400 for a. Subtract. = 85 Simplify.
Evaluate a Piecewise-Defined Function According to this model, the average price per square foot for a home with a square footage of 1400 square feet is $85. Answer:
Evaluate a Piecewise-Defined Function According to this model, the average price per square foot for a home with a square footage of 1400 square feet is $85. Answer: $85 per square foot
Evaluate a Piecewise-Defined Function B. FINANCE Realtors in a metropolitan area studied the average home price per square foot as a function of total square footage. Their evaluation yielded the following piecewise-defined function. Find the average price per square foot for a home with the square footage of 3200 square feet.
Evaluate a Piecewise-Defined Function Because 3200 is between 2600 and 4000, use to find p(3200). Function for 2600 ≤ a < 4000. Substitute 3200 for a. Simplify.
Evaluate a Piecewise-Defined Function According to this model, the average price per square foot for a home with a square footage of 3200 square feet is $104. Answer:
Evaluate a Piecewise-Defined Function According to this model, the average price per square foot for a home with a square footage of 3200 square feet is $104. Answer: $104 per square foot
ENERGY The cost of residential electricity use can be represented by the following piecewise function, where k is the number of kilowatts. Find the cost of electricity for 950 kilowatts. A. $47. 50 B. $48. 00 C. $57. 50 D. $76. 50
ENERGY The cost of residential electricity use can be represented by the following piecewise function, where k is the number of kilowatts. Find the cost of electricity for 950 kilowatts. A. $47. 50 B. $48. 00 C. $57. 50 D. $76. 50
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