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

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

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

4. 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

5. Key Concept 1

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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

13. 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

14. 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

15. 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

16. Key Concept 3

17. Key Concept 3a

18. 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

19. 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

20. 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

21. 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

22. 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

23. 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

24. 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

25. 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

26. 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

27. 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

28. 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

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

30. 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

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

32. 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

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

34. 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

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

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

37. 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 square feet.
Example 6

38. 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

39. 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

40. 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

41. 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