Evaluate |x – 2y| – |2x – y| – xy if x = –2 and y = 7. B. 9 C. 19 D. 41 5–Minute Check 3

Evaluate |x – 2y| – |2x – y| – xy if x = –2 and y = 7. B. 9 C. 19 D. 41 5–Minute Check 3

Describe subsets of real numbers. 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. Then/Now

piecewise-defined function relevant domain set-builder notation interval notation function function notation independent variable dependent variable implied domain piecewise-defined function relevant domain Vocabulary

Key Concept 1

A. Describe {2, 3, 4, 5, 6, 7} using set-builder notation. 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. Example 1

A. Describe {2, 3, 4, 5, 6, 7} using set-builder notation. 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. Example 1

C. Describe all multiples of seven using set-builder notation. Use Set-Builder Notation C. Describe all multiples of seven using set-builder notation. The set includes all integers that are multiples of 7. Example 1

C. Describe all multiples of seven using set-builder notation. Use Set-Builder Notation C. Describe all multiples of seven using set-builder notation. The set includes all integers that are multiples of 7. Example 1

A. Write –2 ≤ x ≤ 12 using interval notation. 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: Example 2

A. Write –2 ≤ x ≤ 12 using interval notation. 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] Example 2

B. Write x > –4 using interval notation. Use Interval Notation B. Write x > –4 using interval notation. The set includes all real numbers greater than –4. Answer: Example 2

B. Write x > –4 using interval notation. Use Interval Notation B. Write x > –4 using interval notation. The set includes all real numbers greater than –4. Answer: (–4, ) Example 2

C. Write x < 3 or x ≥ 54 using interval notation. 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: Example 2

C. Write x < 3 or x ≥ 54 using interval notation. 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: Example 2

Key Concept 3

Key Concept 3a

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: Example 3

Answer: No; there is more than one y-value for an x-value. 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. Example 3

B. Determine whether the table represents y as a function of x. Identify Relations that are Functions B. Determine whether the table represents y as a function of x. Answer: Example 3

B. Determine whether the table represents y as a function of x. 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. Example 3

D. Determine whether x = 3y 2 represents y as a function of x. Identify Relations that are Functions D. Determine whether x = 3y 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 = 3y 2 Original equation Divide each side by 3. Take the square root of each side. Example 3

Determine whether 12x 2 + 4y = 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. Example 3

Determine whether 12x 2 + 4y = 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. Example 3

B. If f (x) = x 2 – 2x – 8, find f (–3d). Find Function Values B. If f (x) = x 2 – 2x – 8, find f (–3d). To find f (–3d), replace x with –3d in f (x) = x 2 – 2x – 8. f (x) = x 2 – 2x – 8 Original function f (–3d) = (–3d)2 – 2(–3d) – 8 Substitute –3d for x. = 9d 2 + 6d – 8 Simplify. Answer: Example 4

B. If f (x) = x 2 – 2x – 8, find f (–3d). Find Function Values B. If f (x) = x 2 – 2x – 8, find f (–3d). To find f (–3d), replace x with –3d in f (x) = x 2 – 2x – 8. f (x) = x 2 – 2x – 8 Original function f (–3d) = (–3d)2 – 2(–3d) – 8 Substitute –3d for x. = 9d 2 + 6d – 8 Simplify. Answer: 9d 2 + 6d – 8 Example 4

C. If f (x) = x2 – 2x – 8, find f (2a – 1). Find Function Values C. If f (x) = x2 – 2x – 8, find f (2a – 1). To find f (2a – 1), replace x with 2a – 1 in f (x) = x 2 – 2x – 8. f (x) = x 2 – 2x – 8 Original function f (2a – 1) = (2a – 1)2 – 2(2a – 1) – 8 Substitute 2a – 1 for x. = 4a 2 – 4a + 1 – 4a + 2 – 8 Expand (2a – 1)2 and 2(2a – 1). = 4a 2 – 8a – 5 Simplify. Answer: Example 4

C. If f (x) = x2 – 2x – 8, find f (2a – 1). Find Function Values C. If f (x) = x2 – 2x – 8, find f (2a – 1). To find f (2a – 1), replace x with 2a – 1 in f (x) = x 2 – 2x – 8. f (x) = x 2 – 2x – 8 Original function f (2a – 1) = (2a – 1)2 – 2(2a – 1) – 8 Substitute 2a – 1 for x. = 4a 2 – 4a + 1 – 4a + 2 – 8 Expand (2a – 1)2 and 2(2a – 1). = 4a 2 – 8a – 5 Simplify. Answer: 4a 2 – 8a – 5 Example 4

A. State the domain of the function . Find Domains Algebraically A. State the domain of the function . Answer: Example 5

A. State the domain of the function . Find Domains Algebraically A. State the domain of the function . Because the square root of a negative number cannot be real, 4x – 1 ≥ 0. Therefore, the domain of g(x) is all real numbers x such that x ≥ , or . Example 5

B. State the domain of the function . Find Domains Algebraically B. State the domain of the function . Example 5

B. State the domain of the function . 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 . Example 5

C. State the domain of the function . Find Domains Algebraically C. State the domain of the function . Example 5

C. State the domain of the function . Find Domains Algebraically C. State the domain of the function . This function is defined only when 2x – 3 > 0. Therefore, the domain of f (x) is or . Example 5

State the domain of g (x) = . A. or [4, ∞) B. or [–4, 4] C. or (− , −4] D. Example 5

State the domain of g (x) = . A. or [4, ∞) B. or [–4, 4] C. or (− , −4] D. Example 5

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 piecewise-defined function. Find the average price per square foot for a home with the square footage of 1400 square feet. Example 6

Because 1400 is between 1000 and 2600, use to find p(1400). 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. Example 6

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 Example 6

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 Example 6

piecewise-defined function relevant domain set-builder notation interval notation function function notation independent variable dependent variable implied domain piecewise-defined function relevant domain Vocabulary