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AA 1.2

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sfulkerson

This document provides an introduction to functions and key function concepts like domain, range, independent and dependent variables. It begins with examples of identifying independent and dependent variables in equations and tables. It then defines a function and discusses how to determine if a set of ordered pairs forms a function. The document concludes by discussing the domain and range of functions.

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WARM-UP: 1. [18 ÷ (2 + 1)] − 4 2. What is the difference of 10 and 1? 3. What is the quotient of 35 and 5?
WARM-UP: 1. [18 ÷ (2 + 1)] − 4 [18 ÷ 3] − 4 € 2. What is the difference of 10 and 1? 3. What is the quotient of 35 and 5?
WARM-UP: 1. [18 ÷ (2 + 1)] − 4 [18 ÷ 3] − 4 [6] − 4 € €2. What is the difference of 10 and 1? 3. What is the quotient of 35 and 5?
WARM-UP: 1. [18 ÷ (2 + 1)] − 4 [18 ÷ 3] − 4 [6] − 4 2 € €2. What is the difference of 10 and 1? € 3. What is the quotient of 35 and 5?
WARM-UP: 1. [18 ÷ (2 + 1)] − 4 [18 ÷ 3] − 4 [6] − 4 2 € €2. What is the difference of 10 and 1? 10 −1 = 9 € 3. What is the quotient of 35 and 5? €
WARM-UP: 1. [18 ÷ (2 + 1)] − 4 [18 ÷ 3] − 4 [6] − 4 2 € €2. What is the difference of 10 and 1? 10 −1 = 9 € 3. What is the quotient of 35 and 5? € 35 ÷ 5 = 7
1.2 WHAT IS A FUNCTION?
ESSENTIAL QUESTION: How do we determine if a set of ordered pairs or table is a function?
VOCABULARY: Dependent Variable: Independent Variable: Function:
VOCABULARY: Dependent Variable: relies on another variable Independent Variable: Function:
VOCABULARY: Dependent Variable: relies on another variable usually “y” variable Independent Variable: Function:
VOCABULARY: Dependent Variable: relies on another variable usually “y” variable Independent Variable: does not rely on another variable Function:
VOCABULARY: Dependent Variable: relies on another variable usually “y” variable Independent Variable: does not rely on another variable usually the “x” variable Function:
VOCABULARY: Dependent Variable: relies on another variable usually “y” variable Independent Variable: does not rely on another variable usually the “x” variable Function: correspondence or pairing between two variables such that each value of the 1st (independent) variable corresponds to exactly one value of the 2nd (dependent) variable
EXAMPLE 1. The equation h = 2t gives the number of inches h of new snow after t hours if snow falls during a storm at the rate of 2 inches per hour. Identify the independent and dependent variables. Independent Variable: Dependent Variable:
EXAMPLE 1. The equation h = 2t gives the number of inches h of new snow after t hours if snow falls during a storm at the rate of 2 inches per hour. Identify the independent and dependent variables. Independent Variable: t - time in hours Dependent Variable:
EXAMPLE 1. The equation h = 2t gives the number of inches h of new snow after t hours if snow falls during a storm at the rate of 2 inches per hour. Identify the independent and dependent variables. Independent Variable: t - time in hours Dependent Variable: h - inches of snow
The dependent variable ______________ the independent variable.
The dependent variable ______________ the independent variable. is a function of
The dependent variable ______________ the independent variable. is a function of y
The dependent variable ______________ the independent variable. is a function of y x
The dependent variable ______________ the independent variable. is a function of y x In your graphing calculator type y = 3x + 7
The dependent variable ______________ the independent variable. is a function of y x In your graphing calculator type y = 3x + 7 Look at the table of values. What is the input and what is the output?
The dependent variable ______________ the independent variable. is a function of y x In your graphing calculator type y = 3x + 7 Look at the table of values. What is the input and what is the output? Input - values for x
The dependent variable ______________ the independent variable. is a function of y x In your graphing calculator type y = 3x + 7 Look at the table of values. What is the input and what is the output? Input - values for x Output - values for y
The table shows the average temperature T in degrees Fahrenheit for each month M in Honolulu, Hawaii. M T Jan. 73 2. Is T a function of M? Feb. 73 March 74 April 76 3. Is M a function of T? May 78 June 79 July 80 August 81 Sept. 81 Oct. 80 Nov. 77 Dec. 74
The table shows the average temperature T in degrees Fahrenheit for each month M in Honolulu, Hawaii. M T Jan. 73 2. Is T a function of M? YES Feb. 73 March 74 April 76 3. Is M a function of T? May 78 June 79 July 80 August 81 Sept. 81 Oct. 80 Nov. 77 Dec. 74
The table shows the average temperature T in degrees Fahrenheit for each month M in Honolulu, Hawaii. M T Jan. 73 2. Is T a function of M? YES Feb. 73 March 74 April 76 3. Is M a function of T? NO May 78 June 79 July 80 August 81 Sept. 81 Oct. 80 Nov. 77 Dec. 74
The table shows the average temperature T in degrees Fahrenheit for each month M in Honolulu, Hawaii. M T Jan. 73 2. Is T a function of M? YES Feb. 73 March 74 April 76 3. Is M a function of T? NO May 78 June 79 July 80 August 81 What is the difference between Sept. 81 Oct. 80 the 2 questions? Nov. 77 Dec. 74
The table shows the average temperature T in degrees Fahrenheit for each month M in Honolulu, Hawaii. M T Jan. 73 2. Is T a function of M? YES Feb. 73 March 74 April 76 3. Is M a function of T? NO May 78 June 79 July 80 August 81 What is the difference between Sept. 81 Oct. 80 the 2 questions? Nov. 77 Dec. 74 According to the definition; the 1st variable can only correspond to 1 value of the 2nd variable. ie the second variable can not be listed twice.
EXAMPLE: s 1 1 2 2 3 3 r 3 -3 6 -6 9 -9 4. Is r a function of s?
EXAMPLE: s 1 1 2 2 3 3 r 3 -3 6 -6 9 -9 4. Is r a function of s? No; each s-value is not paired with exactly 1 r-value OR s has repeated values
EXAMPLE: s 1 1 2 2 3 3 r 3 -3 6 -6 9 -9 4. Is r a function of s? No; each s-value is not paired with exactly 1 r-value OR s has repeated values 5. Is s a function of r?
EXAMPLE: s 1 1 2 2 3 3 r 3 -3 6 -6 9 -9 4. Is r a function of s? No; each s-value is not paired with exactly 1 r-value OR s has repeated values 5. Is s a function of r? Yes; every r-value is paired with exactly 1 s-value OR r does not have repeated values
EXAMPLE: 6. The table gives the high school enrollment, in millions, in the United States from 1985 to 1991. Is the female enrollment a function of the year? Year Male Female 1985 7.2 6.9 1986 7.2 7.0 1987 7.0 6.8 1988 6.7 6.4 1989 6.6 6.3 1990 6.5 6.4 1991 6.8 6.4
EXAMPLE: 6. The table gives the high school enrollment, in millions, in the United States from 1985 to 1991. Is the female enrollment a function of the year? Year Male Female 1985 7.2 6.9 Yes; 1986 7.2 7.0 each year is paired with exactly 1 female enrollment figure 1987 7.0 6.8 OR 1988 6.7 6.4 the year does not repeat itself 1989 6.6 6.3 1990 6.5 6.4 1991 6.8 6.4
VOCABULARY CONTINUED... Domain of a Function: Range of a Function:
VOCABULARY CONTINUED... Domain of a Function: set of values, which are allowable substitutions for the independent variable (x-values) (INPUT) Range of a Function:
VOCABULARY CONTINUED... Domain of a Function: set of values, which are allowable substitutions for the independent variable (x-values) (INPUT) Range of a Function: set of values of the dependent variable that can result from the substitution for the independent variable (y-values) (OUTPUT)
VOCABULARY CONTINUED... Domain of a Function: set of values, which are allowable substitutions for the independent variable (x-values) (INPUT) Range of a Function: set of values of the dependent variable that can result from the substitution for the independent variable (y-values) (OUTPUT) Refer to the temperature example:
VOCABULARY CONTINUED... Domain of a Function: set of values, which are allowable substitutions for the independent variable (x-values) (INPUT) Range of a Function: set of values of the dependent variable that can result from the substitution for the independent variable (y-values) (OUTPUT) Refer to the temperature example: Domain:
VOCABULARY CONTINUED... Domain of a Function: set of values, which are allowable substitutions for the independent variable (x-values) (INPUT) Range of a Function: set of values of the dependent variable that can result from the substitution for the independent variable (y-values) (OUTPUT) Refer to the temperature example: Domain: Range:
VOCABULARY CONTINUED... Domain of a Function: set of values, which are allowable substitutions for the independent variable (x-values) (INPUT) Range of a Function: set of values of the dependent variable that can result from the substitution for the independent variable (y-values) (OUTPUT) Refer to the temperature example: Domain: set of months in a year Range:
VOCABULARY CONTINUED... Domain of a Function: set of values, which are allowable substitutions for the independent variable (x-values) (INPUT) Range of a Function: set of values of the dependent variable that can result from the substitution for the independent variable (y-values) (OUTPUT) Refer to the temperature example: Domain: set of months in a year Range: {73, 74, 76, 77, 78, 79, 80, 81}
EXAMPLE: 7. If y is a function of x, what real numbers are not in the 1 domain of y = 2 ? x − 64 €
EXAMPLE: 7. If y is a function of x, what real numbers are not in the 1 domain of y = 2 ? x − 64 Clue: Is there a value for x that I can not have? €
EXAMPLE: 7. If y is a function of x, what real numbers are not in the 1 domain of y = 2 ? x − 64 Clue: Is there a value for x that I can not have? 8 €
EXAMPLE: 7. If y is a function of x, what real numbers are not in the 1 domain of y = 2 ? x − 64 Clue: Is there a value for x that I can not have? 8 € and
EXAMPLE: 7. If y is a function of x, what real numbers are not in the 1 domain of y = 2 ? x − 64 Clue: Is there a value for x that I can not have? 8 € and -8
EXAMPLE: 7. If y is a function of x, what real numbers are not in the 1 domain of y = 2 ? x − 64 Clue: Is there a value for x that I can not have? 8 € and -8 If x is 8 or -8 then the denominator is 0. Everyone knows we can’t divide by 0. Therefore we can not have 8 and -8 as a value for x.
8. What is the domain and range of {(2, 4), (7, 11), (9, 13), (8, -4)} Domain: Range:
8. What is the domain and range of {(2, 4), (7, 11), (9, 13), (8, -4)} Domain: {2, 7, 8, 9} Range:
8. What is the domain and range of {(2, 4), (7, 11), (9, 13), (8, -4)} Domain: {2, 7, 8, 9} Range: {-4, 4, 11, 13}
8. What is the domain and range of {(2, 4), (7, 11), (9, 13), (8, -4)} Domain: {2, 7, 8, 9} Range: {-4, 4, 11, 13} Notice: the numbers are listed in ascending order
8. What is the domain and range of {(2, 4), (7, 11), (9, 13), (8, -4)} Domain: {2, 7, 8, 9} Range: {-4, 4, 11, 13} Notice: the numbers are listed in ascending order 9. What is the domain and range of y = x4 - 3
8. What is the domain and range of {(2, 4), (7, 11), (9, 13), (8, -4)} Domain: {2, 7, 8, 9} Range: {-4, 4, 11, 13} Notice: the numbers are listed in ascending order 9. What is the domain and range of y = x4 - 3 For this let’s examine the graph
8. What is the domain and range of {(2, 4), (7, 11), (9, 13), (8, -4)} Domain: {2, 7, 8, 9} Range: {-4, 4, 11, 13} Notice: the numbers are listed in ascending order 9. What is the domain and range of y = x4 - 3 For this let’s examine the graph
8. What is the domain and range of {(2, 4), (7, 11), (9, 13), (8, -4)} Domain: {2, 7, 8, 9} Range: {-4, 4, 11, 13} Notice: the numbers are listed in ascending order 9. What is the domain and range of y = x4 - 3 For this let’s examine the graph Domain:
8. What is the domain and range of {(2, 4), (7, 11), (9, 13), (8, -4)} Domain: {2, 7, 8, 9} Range: {-4, 4, 11, 13} Notice: the numbers are listed in ascending order 9. What is the domain and range of y = x4 - 3 For this let’s examine the graph Domain: all real numbers
8. What is the domain and range of {(2, 4), (7, 11), (9, 13), (8, -4)} Domain: {2, 7, 8, 9} Range: {-4, 4, 11, 13} Notice: the numbers are listed in ascending order 9. What is the domain and range of y = x4 - 3 For this let’s examine the graph Domain: all real numbers Range:
8. What is the domain and range of {(2, 4), (7, 11), (9, 13), (8, -4)} Domain: {2, 7, 8, 9} Range: {-4, 4, 11, 13} Notice: the numbers are listed in ascending order 9. What is the domain and range of y = x4 - 3 For this let’s examine the graph Domain: all real numbers Range: {y : y ≥ −3}
SETS OF NUMBERS * often used for the domains Natural Numbers: Whole Numbers: Integers: Rational Numbers: Real Numbers:
SETS OF NUMBERS * often used for the domains Natural Numbers: a.k.a. ~ counting numbers Whole Numbers: Integers: Rational Numbers: Real Numbers:
SETS OF NUMBERS * often used for the domains Natural Numbers: a.k.a. ~ counting numbers {1, 2, 3, 4, 5, 6, ...} Whole Numbers: Integers: Rational Numbers: Real Numbers:
SETS OF NUMBERS * often used for the domains Natural Numbers: a.k.a. ~ counting numbers {1, 2, 3, 4, 5, 6, ...} Whole Numbers: {0, 1, 2, 3, 4, 5,...} Integers: Rational Numbers: Real Numbers:
SETS OF NUMBERS * often used for the domains Natural Numbers: a.k.a. ~ counting numbers {1, 2, 3, 4, 5, 6, ...} Whole Numbers: {0, 1, 2, 3, 4, 5,...} Integers: {...-3, -2, -1, 0, 1, 2, 3,...} Rational Numbers: Real Numbers:
SETS OF NUMBERS * often used for the domains Natural Numbers: a.k.a. ~ counting numbers {1, 2, 3, 4, 5, 6, ...} Whole Numbers: {0, 1, 2, 3, 4, 5,...} Integers: {...-3, -2, -1, 0, 1, 2, 3,...} Rational Numbers: numbers that can be represented as a ratio; a/b where b can’t be 0. Real Numbers:
SETS OF NUMBERS * often used for the domains Natural Numbers: a.k.a. ~ counting numbers {1, 2, 3, 4, 5, 6, ...} Whole Numbers: {0, 1, 2, 3, 4, 5,...} Integers: {...-3, -2, -1, 0, 1, 2, 3,...} Rational Numbers: numbers that can be represented as a ratio; a/b where b can’t be 0. Real Numbers: set of numbers represented by decimals (all numbers known to YOU currently 0, -7.2, pi
EXAMPLES: Give an example that will satisfy each of the conditions. 10. an integer that is not a natural number
EXAMPLES: Give an example that will satisfy each of the conditions. 10. an integer that is not a natural number -5 - any negative number
EXAMPLES: Give an example that will satisfy each of the conditions. 10. an integer that is not a natural number -5 - any negative number 11. a real number that is not an integer
EXAMPLES: Give an example that will satisfy each of the conditions. 10. an integer that is not a natural number -5 - any negative number 11. a real number that is not an integer pi, .9 - any fraction
EXAMPLES: Give an example that will satisfy each of the conditions. 10. an integer that is not a natural number -5 - any negative number 11. a real number that is not an integer pi, .9 - any fraction 12. an integer that is not a real number
EXAMPLES: Give an example that will satisfy each of the conditions. 10. an integer that is not a natural number -5 - any negative number 11. a real number that is not an integer pi, .9 - any fraction 12. an integer that is not a real number not possible
HOMEWORK: page 15 #1-14 and 20-31

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AA 1.2

  • 1. WARM-UP: 1. [18 ÷ (2 + 1)] − 4 2. What is the difference of 10 and 1? 3. What is the quotient of 35 and 5?
  • 2. WARM-UP: 1. [18 ÷ (2 + 1)] − 4 [18 ÷ 3] − 4 € 2. What is the difference of 10 and 1? 3. What is the quotient of 35 and 5?
  • 3. WARM-UP: 1. [18 ÷ (2 + 1)] − 4 [18 ÷ 3] − 4 [6] − 4 € €2. What is the difference of 10 and 1? 3. What is the quotient of 35 and 5?
  • 4. WARM-UP: 1. [18 ÷ (2 + 1)] − 4 [18 ÷ 3] − 4 [6] − 4 2 € €2. What is the difference of 10 and 1? € 3. What is the quotient of 35 and 5?
  • 5. WARM-UP: 1. [18 ÷ (2 + 1)] − 4 [18 ÷ 3] − 4 [6] − 4 2 € €2. What is the difference of 10 and 1? 10 −1 = 9 € 3. What is the quotient of 35 and 5? €
  • 6. WARM-UP: 1. [18 ÷ (2 + 1)] − 4 [18 ÷ 3] − 4 [6] − 4 2 € €2. What is the difference of 10 and 1? 10 −1 = 9 € 3. What is the quotient of 35 and 5? € 35 ÷ 5 = 7
  • 7. 1.2 WHAT IS A FUNCTION?
  • 8. ESSENTIAL QUESTION: How do we determine if a set of ordered pairs or table is a function?
  • 10. VOCABULARY: Dependent Variable: relies on another variable Independent Variable: Function:
  • 11. VOCABULARY: Dependent Variable: relies on another variable usually “y” variable Independent Variable: Function:
  • 12. VOCABULARY: Dependent Variable: relies on another variable usually “y” variable Independent Variable: does not rely on another variable Function:
  • 13. VOCABULARY: Dependent Variable: relies on another variable usually “y” variable Independent Variable: does not rely on another variable usually the “x” variable Function:
  • 14. VOCABULARY: Dependent Variable: relies on another variable usually “y” variable Independent Variable: does not rely on another variable usually the “x” variable Function: correspondence or pairing between two variables such that each value of the 1st (independent) variable corresponds to exactly one value of the 2nd (dependent) variable
  • 15. EXAMPLE 1. The equation h = 2t gives the number of inches h of new snow after t hours if snow falls during a storm at the rate of 2 inches per hour. Identify the independent and dependent variables. Independent Variable: Dependent Variable:
  • 16. EXAMPLE 1. The equation h = 2t gives the number of inches h of new snow after t hours if snow falls during a storm at the rate of 2 inches per hour. Identify the independent and dependent variables. Independent Variable: t - time in hours Dependent Variable:
  • 17. EXAMPLE 1. The equation h = 2t gives the number of inches h of new snow after t hours if snow falls during a storm at the rate of 2 inches per hour. Identify the independent and dependent variables. Independent Variable: t - time in hours Dependent Variable: h - inches of snow
  • 18. The dependent variable ______________ the independent variable.
  • 19. The dependent variable ______________ the independent variable. is a function of
  • 20. The dependent variable ______________ the independent variable. is a function of y
  • 21. The dependent variable ______________ the independent variable. is a function of y x
  • 22. The dependent variable ______________ the independent variable. is a function of y x In your graphing calculator type y = 3x + 7
  • 23. The dependent variable ______________ the independent variable. is a function of y x In your graphing calculator type y = 3x + 7 Look at the table of values. What is the input and what is the output?
  • 24. The dependent variable ______________ the independent variable. is a function of y x In your graphing calculator type y = 3x + 7 Look at the table of values. What is the input and what is the output? Input - values for x
  • 25. The dependent variable ______________ the independent variable. is a function of y x In your graphing calculator type y = 3x + 7 Look at the table of values. What is the input and what is the output? Input - values for x Output - values for y
  • 26. The table shows the average temperature T in degrees Fahrenheit for each month M in Honolulu, Hawaii. M T Jan. 73 2. Is T a function of M? Feb. 73 March 74 April 76 3. Is M a function of T? May 78 June 79 July 80 August 81 Sept. 81 Oct. 80 Nov. 77 Dec. 74
  • 27. The table shows the average temperature T in degrees Fahrenheit for each month M in Honolulu, Hawaii. M T Jan. 73 2. Is T a function of M? YES Feb. 73 March 74 April 76 3. Is M a function of T? May 78 June 79 July 80 August 81 Sept. 81 Oct. 80 Nov. 77 Dec. 74
  • 28. The table shows the average temperature T in degrees Fahrenheit for each month M in Honolulu, Hawaii. M T Jan. 73 2. Is T a function of M? YES Feb. 73 March 74 April 76 3. Is M a function of T? NO May 78 June 79 July 80 August 81 Sept. 81 Oct. 80 Nov. 77 Dec. 74
  • 29. The table shows the average temperature T in degrees Fahrenheit for each month M in Honolulu, Hawaii. M T Jan. 73 2. Is T a function of M? YES Feb. 73 March 74 April 76 3. Is M a function of T? NO May 78 June 79 July 80 August 81 What is the difference between Sept. 81 Oct. 80 the 2 questions? Nov. 77 Dec. 74
  • 30. The table shows the average temperature T in degrees Fahrenheit for each month M in Honolulu, Hawaii. M T Jan. 73 2. Is T a function of M? YES Feb. 73 March 74 April 76 3. Is M a function of T? NO May 78 June 79 July 80 August 81 What is the difference between Sept. 81 Oct. 80 the 2 questions? Nov. 77 Dec. 74 According to the definition; the 1st variable can only correspond to 1 value of the 2nd variable. ie the second variable can not be listed twice.
  • 31. EXAMPLE: s 1 1 2 2 3 3 r 3 -3 6 -6 9 -9 4. Is r a function of s?
  • 32. EXAMPLE: s 1 1 2 2 3 3 r 3 -3 6 -6 9 -9 4. Is r a function of s? No; each s-value is not paired with exactly 1 r-value OR s has repeated values
  • 33. EXAMPLE: s 1 1 2 2 3 3 r 3 -3 6 -6 9 -9 4. Is r a function of s? No; each s-value is not paired with exactly 1 r-value OR s has repeated values 5. Is s a function of r?
  • 34. EXAMPLE: s 1 1 2 2 3 3 r 3 -3 6 -6 9 -9 4. Is r a function of s? No; each s-value is not paired with exactly 1 r-value OR s has repeated values 5. Is s a function of r? Yes; every r-value is paired with exactly 1 s-value OR r does not have repeated values
  • 35. EXAMPLE: 6. The table gives the high school enrollment, in millions, in the United States from 1985 to 1991. Is the female enrollment a function of the year? Year Male Female 1985 7.2 6.9 1986 7.2 7.0 1987 7.0 6.8 1988 6.7 6.4 1989 6.6 6.3 1990 6.5 6.4 1991 6.8 6.4
  • 36. EXAMPLE: 6. The table gives the high school enrollment, in millions, in the United States from 1985 to 1991. Is the female enrollment a function of the year? Year Male Female 1985 7.2 6.9 Yes; 1986 7.2 7.0 each year is paired with exactly 1 female enrollment figure 1987 7.0 6.8 OR 1988 6.7 6.4 the year does not repeat itself 1989 6.6 6.3 1990 6.5 6.4 1991 6.8 6.4
  • 37. VOCABULARY CONTINUED... Domain of a Function: Range of a Function:
  • 38. VOCABULARY CONTINUED... Domain of a Function: set of values, which are allowable substitutions for the independent variable (x-values) (INPUT) Range of a Function:
  • 39. VOCABULARY CONTINUED... Domain of a Function: set of values, which are allowable substitutions for the independent variable (x-values) (INPUT) Range of a Function: set of values of the dependent variable that can result from the substitution for the independent variable (y-values) (OUTPUT)
  • 40. VOCABULARY CONTINUED... Domain of a Function: set of values, which are allowable substitutions for the independent variable (x-values) (INPUT) Range of a Function: set of values of the dependent variable that can result from the substitution for the independent variable (y-values) (OUTPUT) Refer to the temperature example:
  • 41. VOCABULARY CONTINUED... Domain of a Function: set of values, which are allowable substitutions for the independent variable (x-values) (INPUT) Range of a Function: set of values of the dependent variable that can result from the substitution for the independent variable (y-values) (OUTPUT) Refer to the temperature example: Domain:
  • 42. VOCABULARY CONTINUED... Domain of a Function: set of values, which are allowable substitutions for the independent variable (x-values) (INPUT) Range of a Function: set of values of the dependent variable that can result from the substitution for the independent variable (y-values) (OUTPUT) Refer to the temperature example: Domain: Range:
  • 43. VOCABULARY CONTINUED... Domain of a Function: set of values, which are allowable substitutions for the independent variable (x-values) (INPUT) Range of a Function: set of values of the dependent variable that can result from the substitution for the independent variable (y-values) (OUTPUT) Refer to the temperature example: Domain: set of months in a year Range:
  • 44. VOCABULARY CONTINUED... Domain of a Function: set of values, which are allowable substitutions for the independent variable (x-values) (INPUT) Range of a Function: set of values of the dependent variable that can result from the substitution for the independent variable (y-values) (OUTPUT) Refer to the temperature example: Domain: set of months in a year Range: {73, 74, 76, 77, 78, 79, 80, 81}
  • 45. EXAMPLE: 7. If y is a function of x, what real numbers are not in the 1 domain of y = 2 ? x − 64 €
  • 46. EXAMPLE: 7. If y is a function of x, what real numbers are not in the 1 domain of y = 2 ? x − 64 Clue: Is there a value for x that I can not have? €
  • 47. EXAMPLE: 7. If y is a function of x, what real numbers are not in the 1 domain of y = 2 ? x − 64 Clue: Is there a value for x that I can not have? 8 €
  • 48. EXAMPLE: 7. If y is a function of x, what real numbers are not in the 1 domain of y = 2 ? x − 64 Clue: Is there a value for x that I can not have? 8 € and
  • 49. EXAMPLE: 7. If y is a function of x, what real numbers are not in the 1 domain of y = 2 ? x − 64 Clue: Is there a value for x that I can not have? 8 € and -8
  • 50. EXAMPLE: 7. If y is a function of x, what real numbers are not in the 1 domain of y = 2 ? x − 64 Clue: Is there a value for x that I can not have? 8 € and -8 If x is 8 or -8 then the denominator is 0. Everyone knows we can’t divide by 0. Therefore we can not have 8 and -8 as a value for x.
  • 51. 8. What is the domain and range of {(2, 4), (7, 11), (9, 13), (8, -4)} Domain: Range:
  • 52. 8. What is the domain and range of {(2, 4), (7, 11), (9, 13), (8, -4)} Domain: {2, 7, 8, 9} Range:
  • 53. 8. What is the domain and range of {(2, 4), (7, 11), (9, 13), (8, -4)} Domain: {2, 7, 8, 9} Range: {-4, 4, 11, 13}
  • 54. 8. What is the domain and range of {(2, 4), (7, 11), (9, 13), (8, -4)} Domain: {2, 7, 8, 9} Range: {-4, 4, 11, 13} Notice: the numbers are listed in ascending order
  • 55. 8. What is the domain and range of {(2, 4), (7, 11), (9, 13), (8, -4)} Domain: {2, 7, 8, 9} Range: {-4, 4, 11, 13} Notice: the numbers are listed in ascending order 9. What is the domain and range of y = x4 - 3
  • 56. 8. What is the domain and range of {(2, 4), (7, 11), (9, 13), (8, -4)} Domain: {2, 7, 8, 9} Range: {-4, 4, 11, 13} Notice: the numbers are listed in ascending order 9. What is the domain and range of y = x4 - 3 For this let’s examine the graph
  • 57. 8. What is the domain and range of {(2, 4), (7, 11), (9, 13), (8, -4)} Domain: {2, 7, 8, 9} Range: {-4, 4, 11, 13} Notice: the numbers are listed in ascending order 9. What is the domain and range of y = x4 - 3 For this let’s examine the graph
  • 58. 8. What is the domain and range of {(2, 4), (7, 11), (9, 13), (8, -4)} Domain: {2, 7, 8, 9} Range: {-4, 4, 11, 13} Notice: the numbers are listed in ascending order 9. What is the domain and range of y = x4 - 3 For this let’s examine the graph Domain:
  • 59. 8. What is the domain and range of {(2, 4), (7, 11), (9, 13), (8, -4)} Domain: {2, 7, 8, 9} Range: {-4, 4, 11, 13} Notice: the numbers are listed in ascending order 9. What is the domain and range of y = x4 - 3 For this let’s examine the graph Domain: all real numbers
  • 60. 8. What is the domain and range of {(2, 4), (7, 11), (9, 13), (8, -4)} Domain: {2, 7, 8, 9} Range: {-4, 4, 11, 13} Notice: the numbers are listed in ascending order 9. What is the domain and range of y = x4 - 3 For this let’s examine the graph Domain: all real numbers Range:
  • 61. 8. What is the domain and range of {(2, 4), (7, 11), (9, 13), (8, -4)} Domain: {2, 7, 8, 9} Range: {-4, 4, 11, 13} Notice: the numbers are listed in ascending order 9. What is the domain and range of y = x4 - 3 For this let’s examine the graph Domain: all real numbers Range: {y : y ≥ −3}
  • 62. SETS OF NUMBERS * often used for the domains Natural Numbers: Whole Numbers: Integers: Rational Numbers: Real Numbers:
  • 63. SETS OF NUMBERS * often used for the domains Natural Numbers: a.k.a. ~ counting numbers Whole Numbers: Integers: Rational Numbers: Real Numbers:
  • 64. SETS OF NUMBERS * often used for the domains Natural Numbers: a.k.a. ~ counting numbers {1, 2, 3, 4, 5, 6, ...} Whole Numbers: Integers: Rational Numbers: Real Numbers:
  • 65. SETS OF NUMBERS * often used for the domains Natural Numbers: a.k.a. ~ counting numbers {1, 2, 3, 4, 5, 6, ...} Whole Numbers: {0, 1, 2, 3, 4, 5,...} Integers: Rational Numbers: Real Numbers:
  • 66. SETS OF NUMBERS * often used for the domains Natural Numbers: a.k.a. ~ counting numbers {1, 2, 3, 4, 5, 6, ...} Whole Numbers: {0, 1, 2, 3, 4, 5,...} Integers: {...-3, -2, -1, 0, 1, 2, 3,...} Rational Numbers: Real Numbers:
  • 67. SETS OF NUMBERS * often used for the domains Natural Numbers: a.k.a. ~ counting numbers {1, 2, 3, 4, 5, 6, ...} Whole Numbers: {0, 1, 2, 3, 4, 5,...} Integers: {...-3, -2, -1, 0, 1, 2, 3,...} Rational Numbers: numbers that can be represented as a ratio; a/b where b can’t be 0. Real Numbers:
  • 68. SETS OF NUMBERS * often used for the domains Natural Numbers: a.k.a. ~ counting numbers {1, 2, 3, 4, 5, 6, ...} Whole Numbers: {0, 1, 2, 3, 4, 5,...} Integers: {...-3, -2, -1, 0, 1, 2, 3,...} Rational Numbers: numbers that can be represented as a ratio; a/b where b can’t be 0. Real Numbers: set of numbers represented by decimals (all numbers known to YOU currently 0, -7.2, pi
  • 69. EXAMPLES: Give an example that will satisfy each of the conditions. 10. an integer that is not a natural number
  • 70. EXAMPLES: Give an example that will satisfy each of the conditions. 10. an integer that is not a natural number -5 - any negative number
  • 71. EXAMPLES: Give an example that will satisfy each of the conditions. 10. an integer that is not a natural number -5 - any negative number 11. a real number that is not an integer
  • 72. EXAMPLES: Give an example that will satisfy each of the conditions. 10. an integer that is not a natural number -5 - any negative number 11. a real number that is not an integer pi, .9 - any fraction
  • 73. EXAMPLES: Give an example that will satisfy each of the conditions. 10. an integer that is not a natural number -5 - any negative number 11. a real number that is not an integer pi, .9 - any fraction 12. an integer that is not a real number
  • 74. EXAMPLES: Give an example that will satisfy each of the conditions. 10. an integer that is not a natural number -5 - any negative number 11. a real number that is not an integer pi, .9 - any fraction 12. an integer that is not a real number not possible