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Mathematics LibreTexts

3.5E: Exercises

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    30832
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    Practice Makes Perfect

    Find the Domain and Range of a Relation

    In the following exercises, for each relation a. find the domain of the relation b. find the range of the relation.

    1. \({\{(1,4),(2,8),(3,12),(4,16),(5,20)}\}\)

    Answer

    a. \({\{1, 2, 3, 4, 5}\}\) b. \({\{4, 8, 12, 16, 20}\}\)

    2. \({\{(1,−2),(2,−4),(3,−6),(4,−8),(5,−10)}\}\)

    3. \({\{(1,7),(5,3),(7,9),(−2,−3),(−2,8)}\}\)

    Answer

    a. \({\{1, 5, 7, −2}\}\) b. \({\{7, 3, 9, −3, 8}\}\)

    4. \({\{(11,3),(−2,−7),(4,−8),(4,17),(−6,9)}\}\)

    In the following exercises, use the mapping of the relation to a. list the ordered pairs of the relation, b. find the domain of the relation, and c. find the range of the relation.

    5.
    This figure shows two table that each have one column. The table on the left has the header “Name” and lists the names “Rebecca”, “Jennifer”, “John”, “Hector”, “Luis”, “Ebony”, “Raphael”, “Meredith”, “Karen”, and “Joseph”. The table on the right has the header “Birthday” and lists the dates “January 18”, “February 15”, “April 1”, “April 7”, “June 23”, “July 30”, “August 19”, and “November 6”. There are arrows starting at names in the Name table and pointing towards dates in the Birthday table. The first arrow goes from Rebecca to January 18. The second arrow goes from Jennifer to April 1. The third arrow goes from John to January 18. The fourth arrow goes from Hector to June 23. The fifth arrow goes from Luis to February 15. The sixth arrow goes from Ebony to April 7. The seventh arrow goes from Raphael to November 6. The eighth arrow goes from Meredith to August 19. The ninth arrow goes from Karen to August 19. The tenth arrow goes from Joseph to July 30.

    Answer

    a. (Rebecca, January 18), (Jennifer, April 1), (John, January 18), (Hector, June 23), (Luis, February 15), (Ebony, April 7), (Raphael, November 6), (Meredith, August 19), (Karen, August 19), (Joseph, July 30)
    b. {Rebecca, Jennifer, John, Hector, Luis, Ebony, Raphael, Meredith, Karen, Joseph}
    c. {January 18, April 1, June 23, February 15, April 7, November 6, August 19, July 30}

    6.
    This figure shows two table that each have one column. The table on the left has the header “Name” and lists the names “Amy”, “Carol”, “Devon”, “Harrison”, “Jackson”, “Labron”, “Mason”, “Natalie”, “Paul”, and “Sylvester”. The table on the right has the header “Birthday” and lists the dates “January 5”, “January 7”, “February 14”, “March 1”, “April 7”, “May 30”, “July 20”, “August 1”, “November 13”, and “November 26”. There are arrows starting at names in the Name table and pointing towards dates in the Birthday table. The first arrow goes from Amy to February 14. The second arrow goes from Carol to May 30. The third arrow goes from Devon to January 5. The fourth arrow goes from Harrison to January 7. The fifth arrow goes from Jackson to November 26. The sixth arrow goes from Labron to April 7. The seventh arrow goes from Mason to July 20. The eighth arrow goes from Natalie to March 1. The ninth arrow goes from Paul to August 1. The tenth arrow goes from Sylvester to November 13.

    7. For a woman of height \(5'4''\) the mapping below shows the corresponding Body Mass Index (BMI). The body mass index is a measurement of body fat based on height and weight. A BMI of \(18.5–24.9\) is considered healthy.

    This figure shows two table that each have one column. The table on the left has the header “Weight (lbs)” and lists the numbers plus 100, 110, 120, 130, 140, 150, and 160. The table on the right has the header “BMI” and lists the numbers 18. 9, 22. 3, 17. 2, 24. 0, 25. 7, 20. 6, and 27. 5. There are arrows starting at numbers in the weight table and pointing towards numbers in the BMI table. The first arrow goes from plus 100 to 17. 2. The second arrow goes from 110 to 18. 9. The third arrow goes from 120 to 20. 6. The fourth arrow goes from 130 to 22. 3. The fifth arrow goes from 140 to 24. 0. The sixth arrow goes from 150 to 25. 7. The seventh arrow goes from 160 to 27. 5.

    Answer

    a. \((+100, 17. 2), (110, 18.9), (120, 20.6), (130, 22.3), (140, 24.0), (150, 25.7), (160, 27.5)\) b. \({\{+100, 110, 120, 130, 140, 150, 160,}\}\) c. \({\{17.2, 18.9, 20.6, 22.3, 24.0, 25.7, 27.5}\}\)

    8. For a man of height \(5'11''\) the mapping below shows the corresponding Body Mass Index (BMI). The body mass index is a measurement of body fat based on height and weight. A BMI of \(18.5–24.9\) is considered healthy.

    This figure shows two table that each have one column. The table on the left has the header “Weight (lbs)” and lists the numbers 130, 140, 150, 160, 170, 180, 190, and 200. The table on the right has the header “BMI” and lists the numbers 22. 3, 19. 5, 20. 9, 27. 9, 25. 1, 26. 5, 23. 7, and 18. 1. There are arrows starting at numbers in the weight table and pointing towards numbers in the BMI table. The first arrow goes from 130 to 18. 1. The second arrow goes from 140 to 19. 5. The third arrow goes from 150 to 20. 9. The fourth arrow goes from 160 to 22. 3. The fifth arrow goes from 170 to 23. 7. The sixth arrow goes from 180 to 25. 1. The seventh arrow goes from 190 to 26. 5. The eighth arrow goes from 200 to 27. 9.

    In the following exercises, use the graph of the relation to a. list the ordered pairs of the relation b. find the domain of the relation c. find the range of the relation.

    9.
    The figure shows the graph of some points on the x y-coordinate plane. The x and y-axes run from negative 6 to 6. The points (negative 3, 4), (negative 3, negative 1), (0, negative 3), (2, 3), (4, negative 1), and (4, negative 3).

    Answer

    a. \((2, 3), (4, −3), (−2, −1), (−3, 4), (4, −1), (0, −3)\) b. \({\{−3, −2, 0, 2, 4}\}\)
    c. \({\{−3, −1, 3, 4}\}\)

    10.
    The figure shows the graph of some points on the x y-coordinate plane. The x and y-axes run from negative 6 to 6. The points (negative 3, 4), (negative 3, negative 4), (negative 2, 0), (negative 1, 3), (1, 5), and (4, negative 2).

    11.
    The figure shows the graph of some points on the x y-coordinate plane. The x and y-axes run from negative 6 to 6. The points (negative 1, 4), (negative 1, negative 4), (0, 3), (0, negative 3), (1, 4), and (1, negative 4).

    Answer

    a. \((1, 4), (1, −4), (−1, 4), (−1, −4), (0, 3), (0, −3)\) b. \({\{−1, 0, 1}\}\) c. \({\{−4, −3, 3,4}\}\)

    12.
    The figure shows the graph of some points on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The points (negative 2, negative 6), (negative 2, negative 3), (0, 0), (0. 5, 1. 5), (1, 3), and (3, 6).

    Determine if a Relation is a Function

    In the following exercises, use the set of ordered pairs to a. determine whether the relation is a function, b. find the domain of the relation, and c. find the range of the relation.

    13. \( {\{(−3,9),(−2,4),(−1,1), (0,0),(1,1),(2,4),(3,9)}\}\)

    Answer

    a. yes b. \({\{−3, −2, −1, 0, 1, 2, 3}\}\) c. \({\{9, 4, 1, 0}\}\)

    14. \({\{(9,−3),(4,−2),(1,−1),(0,0),(1,1),(4,2),(9,3)}\}\)

    15. \({\{(−3,27),(−2,8),(−1,1), (0,0),(1,1),(2,8),(3,27)}\}\)

    Answer

    a. yes b. \({\{−3, −2, −1, 0, 1, 2, 3}\}\) c. \({\{0, 1, 8, 27}\}\)

    16. \({\{(−3,−27),(−2,−8),(−1,−1), (0,0),(1,1),(2,8),(3,27)}\}\)

    In the following exercises, use the mapping to a. determine whether the relation is a function, b. find the domain of the function, and c. find the range of the function.

    17.
    This figure shows two table that each have one column. The table on the left has the header “Number” and lists the numbers negative 3, negative 2, negative 1, 0, 1, 2, and 3. The table on the right has the header “Absolute Value” and lists the numbers 0, 1, 2, and 3. There are arrows starting at numbers in the number table and pointing towards numbers in the absolute value table. The first arrow goes from negative 3 to 3. The second arrow goes from negative 2 to 2. The third arrow goes from negative 1 to 1. The fourth arrow goes from 0 to 0. The fifth arrow goes from 1 to 1. The sixth arrow goes from 2 to 2. The seventh arrow goes from 3 to 3.

    Answer

    a. yes b. \({\{−3, −2, −1, 0, 1, 2, 3}\}\) c. \({\{0, 1, 2, 3}\}\)

    18.
    This figure shows two table that each have one column. The table on the left has the header “Number” and lists the numbers negative 3, negative 2, negative 1, 0, 1, 2, and 3. The table on the right has the header “Square” and lists the numbers 0, 1, 4, and 9. There are arrows starting at numbers in the number table and pointing towards numbers in the square table. The first arrow goes from negative 3 to 9. The second arrow goes from negative 2 to 4. The third arrow goes from negative 1 to 1. The fourth arrow goes from 0 to 0. The fifth arrow goes from 1 to 1. The sixth arrow goes from 2 to 4. The seventh arrow goes from 3 to 9.

    19.
    This figure shows two table that each have one column. The table on the left has the header “Name” and lists the names “Jenny”, “R and y”, “Dennis”, “Emily”, and “Raul”. The table on the right has the header “Email” and lists the email addresses RHern and ez@state. edu, JKim@gmail.com, Raul@gmail.com, ESmith@state. edu, DBrown@aol.com, jenny@aol.com, and R and y@gmail.com. There are arrows starting at names in the name table and pointing towards addresses in the email table. The first arrow goes from Jenny to JKim@gmail.com. The second arrow goes from Jenny to jenny@aol.com. The third arrow goes from R and y to R and y@gmail.com. The fourth arrow goes from Dennis to DBrown@aol.com. The fifth arrow goes from Emily to ESmith@state. edu. The sixth arrow goes from Raul to RHern and ez@state. edu. The seventh arrow goes from Raul to Raul@gmail.com.

    Answer

    a. no b. {Jenny, R and y, Dennis, Emily, Raul} c. {RHern and ez@state.edu, JKim@gmail.com, Raul@gmail.com, ESmith@state.edu, DBroen@aol.com, jenny@aol.cvom, R and y@gmail.com}

    20.
    This figure shows two table that each have one column. The table on the left has the header “Name” and lists the names “Jon”, “Rachel”, “Matt”, “Leslie”, “Chris”, “Beth”, and “Liz”. The table on the right has the header “Email” and lists the email addresses chrisg@gmail.com, lizzie@aol.com, jong@gmail.com, mattg@gmail.com, Rachel@state. edu, leslie@aol.com, and bethc@gmail.com. There are arrows starting at names in the name table and pointing towards addresses in the email table. The first arrow goes from Jon to jong@gmail.com. The second arrow goes from Rachel to Rachel@state. edu. The third arrow goes from Matt to mattg@gmail.com. The fourth arrow goes from Leslie to leslie@aol.com. The fifth arrow goes from Chris to chrisg@gmail.com. The sixth arrow goes from Beth to bethc@gmail.com. The seventh arrow goes from Liz to lizzie@aol.com.

    In the following exercises, determine whether each equation is a function.

    21. a. \(2x+y=−3\)
    b. \(y=x^2\)
    c. \(x+y^2=−5\)

    Answer

    a. yes b. yes c. no

    22. a. \(y=3x−5\)
    b. \(y=x^3\)
    c. \(2x+y^2=4\)

    23. a. \(y−3x^3=2\)
    b. \(x+y^2=3\)
    c. \(3x−2y=6\)

    Answer

    a. yes b. no c. yes

    24. a. \(2x−4y=8\)
    b. \(−4=x^2−y\)
    c. \(y^2=−x+5\)

    Find the Value of a Function

    In the following exercises, evaluate the function: a. \(f(2)\) b. \(f(−1)\) c. \(f(a)\).

    25. \(f(x)=5x−3\)

    Answer

    a. \(f(2)=7\) b. \(f(−1)=−8\) c. \(f(a)=5a−3\)

    26. \(f(x)=3x+4\)

    27. \(f(x)=−4x+2\)

    Answer

    a. \(f(2)=−6\) b. \(f(−1)=6\) c. \(f(a)=−4a+2\)

    28. \(f(x)=−6x−3\)

    29. \(f(x)=x^2−x+3\)

    Answer

    a. \(f(2)=5\) b. \(f(−1)=5\)
    c. \(f(a)=a^2−a+3\)

    30. \(f(x)=x^2+x−2\)

    31. \(f(x)=2x^2−x+3\)

    Answer

    a. \(f(2)=9\) b. \(f(−1)=6\)
    c. \(f(a)=2a^2−a+3\)

    32. \(f(x)=3x^2+x−2\)

    In the following exercises, evaluate the function: a. \(g(h^2)\) b. \(g(x+2)\) c. \(g(x)+g(2)\).

    33. \(g(x)=2x+1\)

    Answer

    a. \(g(h^2)=2h^2+1\)
    b. \(g(x+2)=4x+5\)
    c. \(g(x)+g(2)=2x+6\)

    34. \(g(x)=5x−8\)

    35. \(g(x)=−3x−2\)

    Answer

    a. \(g(h^2)=−3h^2−2\)
    b. \(g(x+2)=−3x−8\)
    c. \(g(x)+g(2)=−3x−10\)

    36. \(g(x)=−8x+2\)

    37. \(g(x)=3−x\)

    Answer

    a. \(g(h^2)=3−h^2\)
    b. \(g(x+2)=1−x\)
    c. \(g(x)+g(2)=4−x\)

    38. \(g(x)=7−5x\)

    In the following exercises, evaluate the function.

    39. \(f(x)=3x^2−5x\); \(f(2)\)

    Answer

    2

    40. \(g(x)=4x^2−3x\); \(g(3)\)

    41. \(F(x)=2x^2−3x+1\); \(F(−1)\)

    Answer

    6

    42. \(G(x)=3x^2−5x+2\); \(G(−2)\)

    43. \(h(t)=2|t−5|+4\); \(f(−4)\)

    Answer

    22

    44. \(h(y)=3|y−1|−3\); \(h(−4)\)

    45. \(f(x)=x+2x−1\); \(f(2)\)

    Answer

    4

    46. \(g(x)=x−2x+2\); \(g(4)\)

    In the following exercises, solve.

    47. The number of unwatched shows in Sylvia’s DVR is 85. This number grows by 20 unwatched shows per week. The function \(N(t)=85+20t\) represents the relation between the number of unwatched shows, N, and the time, t, measured in weeks.

    a. Determine the independent and dependent variable.

    b. Find \(N(4)\). Explain what this result means

    Answer

    a. t IND; N DEP
    b. \(N(4)=165\) the number of unwatched shows in Sylvia’s DVR at the fourth week.

    48. Every day a new puzzle is downloaded into Ken’s account. Right now he has 43 puzzles in his account. The function \(N(t)=43+t\) represents the relation between the number of puzzles, N, and the time, t, measured in days.

    a. Determine the independent and dependent variable.

    b. Find \(N(30)\). Explain what this result means.

    49. The daily cost to the printing company to print a book is modeled by the function \(C(x)=3.25x+1500\) where C is the total daily cost and x is the number of books printed.

    a. Determine the independent and dependent variable.

    b. Find \(N(0)\). Explain what this result means.

    c. Find \(N(1000)\). Explain what this result means.

    Answer

    a. x IND; C DEP
    b. \(N(0)=1500\) the daily cost if no books are printed
    c. \(N(1000)=4750\) the daily cost of printing 1000 books

    50. The daily cost to the manufacturing company is modeled by the function \(C(x)=7.25x+2500\) where \(C(x)\) is the total daily cost and x is the number of items manufactured.

    a. Determine the independent and dependent variable.

    b. Find \(C(0)\). Explain what this result means.

    c. Find \(C(1000)\). Explain what this result means.

    Writing Exercises

    51. In your own words, explain the difference between a relation and a function.

    52. In your own words, explain what is meant by domain and range.

    53. Is every relation a function? Is every function a relation?

    54. How do you find the value of a function?

    Self Check

    a. After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

    The figure shows a table with four rows and four columns. The first row is a header row and it labels each column. The first column header is “I can…”, the second is “confidently”, the third is “with some help”, “no minus I don’t get it!”. Under the first column are the phrases “find the domain and range of a relation”, “determine if a relation is a function”, and “find the value of a function”. Under the second, third, fourth columns are blank spaces where the learner can check what level of mastery they have achieved.

    b. After looking at the checklist, do you think you are well-prepared for the next section? Why or why not?