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4.1E: Exercises - Functions and Function Notation

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    147263
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    Practice Makes Perfect

    In the following exercises, use the graph of the relation to:

    1. list the ordered pairs of the relation
    2. determine if the relation is a function
    3. find the domain of the relation
    4. find the range of the relation.

    1.
    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. no

    c. \({\{−3, −2, 0, 2, 4}\}\)

    d. \({\{−3, −1, 3, 4}\}\)

    2.
    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).

    3.
    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. no

    c. \({\{−1, 0, 1}\}\)

    d. \({\{−4, −3, 3,4}\}\)

    4.
    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:

    1. determine if the relation is a function
    2. find the domain of the relation
    3. find the range of the relation.

    5. \( {\{(−3,9),(−2,4),(−1,1), (0,0),(1,1),(2,4),(3,9)}\}\)

    Answer

    a. yes

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

    c. \({\{9, 4, 1, 0}\}\)

    6. \({\{(9,−3),(4,−2),(1,−1),(0,0),(1,1),(4,2),(9,3)}\}\)

    7. \({\{(−3,27),(−2,8),(−1,1), (0,0),(1,1),(2,8),(3,27)}\}\)

    Answer

    a. yes

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

    c. \({\{0, 1, 8, 27}\}\)

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

    In the following exercises, use the mapping to:

    1. determine if the relation is a function
    2. find the domain of the relation
    3. find the range of the relation.

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

    10.
    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.

    11.
    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. {RHernandez@state.edu, JKim@gmail.com, Raul@gmail.com, ESmith@state.edu, DBrown@aol.com, jenny@aol.cvom, Randy@gmail.com}

    12.
    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.

    13.

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

    Answer

    a. yes

    b. yes

    c. no

    14.

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

    15.

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

    Answer

    a. yes

    b. no

    c. yes

    16.

    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. \(f(a)\).

    17. \(f(x)=5x−3\)

    Answer

    a. \(f(2)=7\)

    b. \(f(−1)=−8\)

    c. \(f(a)=5a−3\)

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

    19. \(f(x)=−4x+2\)

    Answer

    a. \(f(2)=−6\)

    b. \(f(−1)=6\)

    c. \(f(a)=−4a+2\)

    20. \(f(x)=−6x−3\)

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

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

    Answer

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

    22. \(g(x)=5x−8\)

    23. \(g(x)=3−x\)

    Answer

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

    24. \(g(x)=7−5x\)

    In the following exercises, evaluate the function.

    25. \(f(x)=3x^2−5x\); \(f(2)\)

    Answer

    2

    26. \(g(x)=4x^2−3x\); \(g(3)\)

    27. \(f(x)=2x^2−3x+1\); \(f(−1)\)

    Answer

    6

    28. \(g(x)=3x^2−5x+2\); \(g(−2)\)

    In the following exercises, solve.

    29. 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.

    30. 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.

    31. 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

    32. 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.

     


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