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2.1E: Exercises

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    Practice Makes Perfect

    Represent Relations as Mapping Diagrams and Tables

    For each of the following relations, create a mapping diagram and table.

    1. \(R = \{(-17, 3), (4, 2), (2, 0), (6, -10), (7, 8)\}\)
    2. \(R = \{(0, 3), (0, -5), (0, 7), (3, 22), (5, 16), (7, 3), (7, -11)\}\)
    3. \(R = \{(-5, -1), (-4, 0), (-3, 1), (-2, 2)\}\)
    4. \(R = \{(-3, 0), (-2, 3), (-2, 4), (1, -5)\}\)
    Answer
    Mapping diagram Table
    exercises_dr_map_1.JPG \(\begin{array} {|c|c|}\hline x & y \\\hline -17 & -10 \\ \hline 2 & 0 \\ \hline 4 & 2 \\ \hline 6 & 3 \\ \hline 7 & 8 \\ \hline \end{array}\)
    1. Mapping diagram Table
      exercises_dr_map_2.JPG \(\begin{array} {|c|c|}\hline x & y \\\hline 0 & -5 \\ \hline 0 & 3\\ \hline 0 & 7 \\ \hline 3 & 22 \\ \hline 5 & 16 \\ \hline 7 & -11 \\ \hline 7 & 3 \\ \hline \end{array}\)
    1. Mapping diagram Table
      exercises_dr_map_3.JPG \(\begin{array} {|c|c|}\hline x & y \\\hline -5 & -1 \\ \hline -4 & 0\\ \hline -3 & 1\\ \hline -2 & 2 \\ \hline \end{array}\)
    1. Mapping diagram Table
      exercises_dr_map_4.JPG \(\begin{array} {|c|c|}\hline x & y \\\hline -3 & 0 \\ \hline -2 & 3\\ \hline -2 & 4\\ \hline 1 & 5 \\ \hline \end{array}\)
    Determine the domain of a relation

    For each relation below, determine the domain.

    1. \(R = \{(-17, 3), (4, 2), (2, 0), (6, -10), (7, 8)\}\)
    2. \(R = \{(0, 3), (0, -5), (0, 7), (3, 22), (5, 16), (7, 3), (7, -11)\}\)
    3. \(R = \{(-5, -1), (-4, 0), (-3, 1), (-2, 2)\}\)
    4. \(R = \{(-3, 0), (-2, 3), (-2, 4), (1, -5)\}\)
    Answer
    1. \(\{-17, 2, 4, 6, 7\}\)
    2. \(\{0, 3, 5, 7\}\)
    3. \(\{-5, -4, -3, -2\}\)
    4. \(\{-3, -2, 1\}\)
    Determine the range of a relation

    For each relation below, determine the range.

    1. \(R = \{(-17, 3), (4, 2), (2, 0), (6, -10), (7, 8)\}\)
    2. \(R = \{(0, 3), (0, -5), (0, 7), (3, 22), (5, 16), (7, 3), (7, -11)\}\)
    3. \(R = \{(-5, -1), (-4, 0), (-3, 1), (-2, 2)\}\)
    4. \(R = \{(-3, 0), (-2, 3), (-2, 4), (1, -5)\}\)
    Answer
    1. \(\{-10, 0, 2, 3, 8\}\)
    2. \(\{-11, -5, 3, 7, 16, 22\}\)
    3. \(\{-1, 0, 1, 2\}\)
    4. \(\{-5, 0, 3, 4\}\)
    Determine the domain and range of a relation

    For each relation below, find the domain and range.

    1. \(R = \{(-13, 7), (2, 4), (0, 0), (10, 6), (8, 7)\}\)
    2. \(R = \{(1, 1), (-1, -4), (-1, 6), (2, 21), (4, 15), (6, 2), (6, -10)\}\)
    3. \(R = \{(-4, 0), (-3, -1), (-4, 0), (-1, 1)\}\)
    4. \(R = \{(-2, 1), (-1, 4), (-3, 5), (0, -5)\}\)
    Answer
    1. Domain: \(\{-13, 0, 2, 8, 10\}\); Range: \(\{0, 4, 6, 7\}\)
    2. Domain: \(\{-1, 1, 2, 4, 6\}\); Range: \(\{-10, -4, 1, 2, 15, 21\}\)
    3. Domain: \(\{-4, -3, -1\}\); Range: \(\{-1, 0, 1\}\)
    4. Domain: \(\{-3, -2, -1, 0\}\); Range: \(\{-5, 1, 4, 5\}\)
    Determine the domain and range of a relation using a table

    For each relation below, determine the domain and range.

    1. \(\begin{array} {|c|c|}\hline x & y \\ \hline -3 & -10 \\ \hline -2 & 3 \\ \hline 0 & -1 \\ \hline 1 & 4 \\ \hline 1 & 6 \\ \hline 5 & 10 \\ \hline 6 & 11 \\ \hline \end{array}\)
    1. \(\begin{array} {|c|c|}\hline x & y \\ \hline -6 & 13 \\ \hline -5 & 31 \\ \hline -3 & -11 \\ \hline -2 & 2 \\ \hline -1 & 8 \\ \hline 0 & -6 \\ \hline 2 & -6 \\ \hline \end{array}\)
    1. \(\begin{array} {|c|c|}\hline x & y \\ \hline -4 & 4 \\ \hline 2 & -2 \\ \hline 3 & -2 \\ \hline 3 & 3 \\ \hline 3 & -3 \\ \hline 10 & -10 \\ \hline 14 & -14 \\ \hline \end{array}\)
    Answer
    1. Domain: \(\{-3, -2, 0, 1, 5, 6\}\); Range: \(\{-10, -1, 3, 4, 6, 10, 11\}\)
    2. Domain: \(\{-6, -5, -3, -2, -1, 0, 2\}\); Range: \(\{-11, -6, 2, 13, 31\}\)
    3. Domain: \(\{-4, 2, 3, 10, 14\}\); Range: \(\{-14, -10, -3, -2, 2, 3, 4\}\)
    Determine if a relation is a function

    Determine if the following relations are functions:

    1. \(R = \{(-13, 7), (2, 4), (0, 0), (10, 6), (8, 7)\}\)
    2. \(R = \{(1, 1), (-1, -4), (-1, 6), (2, 21), (4, 15), (6, 2), (6, -10)\}\)
    3. \(R = \{(-4, 0), (-3, -1), (-4, 1), (-1, 1)\}\)
    4. \(R = \{(-2, 1), (-1, 4), (-3, 5), (0, -5)\}\)
    5. \(R = \{(-17, 3), (4, 2), (2, 0), (6, -10), (7, 8)\}\)
    6. \(R = \{(0, 3), (0, -5), (0, 7), (3, 22), (5, 16), (7, 3), (7, -11)\}\)
    7. \(R = \{(-5, -1), (-4, 0), (-3, 1), (-2, 2)\}\)
    8. \(R = \{(-3, 0), (-2, 3), (-2, 4), (1, -5)\}\)
    Answer
    1. Function
    2. Not a function
    3. Not a function
    4. Function
    5. Function
    6. Not a function
    7. Function
    8. Not a function
    Evaluate a function represented with ordered pairs

    For each function below, evaluate the following: \(f(-1), f(0), f(3)\)

    1. \(f = \{(-1, 3), (0, 5), (3, 16)\}\)
    2. \(f = \{(-4, 5), (-1, 2), (0, 5), (3, 5)\}\)
    3. \(f = \{(-13, 7), (-1, -1), (2, 4), (0, 0), (3, 4) (10, 6), (8, 7)\}\)
    4. \(f = \{(-5, -1), (-4, 0), (-3, 1), (-2, 2), (-1, 12), (0, -4), (3, -2)\}\)
    5. \(f = \{(-3, 0), (-2, 3), (-1, 9), (0, 4), (1, -5), (3, -14), (9, 2)\}\)
    Answer
    1. \(f(-1) = 3; f(0) = 5; f(3) = 16\)
    2. \(f(-1) = 2; f(0) = 5; f(3) = 5\)
    3. \(f(-1) = -1; f(0) = 0, f(3) = 4\)
    4. \(f(-1) = 12; f(0) = -4; f(3) = -2\)
    5. \(f(-1) = 9; f(0) = 4; f(3) = -14\)
    Evaluate functions defined using an equation

    For each function below, evaluate the following: \(f(-3), f(0), f(5)\)

    1. \(f(x) = 2x + 1\)
    2. \(f(x) = x^2 + 1\)
    3. \(f(x) = \sqrt{x + 4}\)
    4. \(f(x) = |x| + 5\)
    5. \(f(x) = -17\)
    6. \(f(x) = 3x - 4\)
    7. \(f(x) = 2x^2 - 2\)
    8. \(f(x) = \sqrt{x + 4} + 9\)
    9. \(f(x) = -2|x| + 3\)
    10. \(f(x) = 3\)
    Answer
    1. \(f(-3) = -5; f(0) = 1, f(5) = 11\)
    2. \(f(-3) = 10; f(0) = 1; f(5) = 26\)
    3. \(f(-3) = 1; f(0) = 2; f(5) = 3\)
    4. \(f(-3) = 8; f(0) = 5; f(5) = 10\)
    5. \(f(-3) = -17; f(0) = -17; f(5) = -17\)
    6. \(f(-3) = -13; f(0) = -4; f(5) = 11\)
    7. \(f(-3) = 16; f(0) = -2; f(5) = 48\)
    8. \(f(-3) = 10; f(0) = 11; f(5) = 12\)
    9. \(f(-3) = -3; f(0) = 3; f(5) = -7\)
    10. \(f(-3) = 3; f(0) = 3; f(5) = 3\)
    Find the domain and range of a function defined using an equation

    Find the domain and range for each of the functions below:

    1. \(f(x) = 2x + 1\)
    2. \(f(x) = x^2 + 1\)
    3. \(f(x) = \sqrt{x + 4}\)
    4. \(f(x) = |x| + 5\)
    5. \(f(x) = -17\)
    6. \(f(x) = 3x - 4\)
    7. \(f(x) = 2x^2 - 2\)
    8. \(f(x) = \sqrt{x + 4} + 9\)
    9. \(f(x) = -2|x| + 3\)
    10. \(f(x) = 3\)
    Answer
    1. Domain: \((-\infty, \infty)\); Range: \((-\infty, \infty)\)
    2. Domain: \((-\infty, \infty)\); Range: \((1, \infty)\)
    3. Domain: \(x \geq -4\); Range: \(y \geq 0\)
    4. Domain: \((-\infty, \infty)\); Range: \(y \geq 5\)
    5. Domain: \((-\infty, \infty)\); Range: \(y = -17\)
    6. Domain: \((-\infty, \infty)\); Range: \((-\infty, \infty)\)
    7. Domain: \((-\infty, \infty)\); Range: \((-2, \infty)\)
    8. Domain: \(x \geq -4\); Range: \(y \geq 9\)
    9. Domain: \((-\infty, \infty)\); Range: \(y \leq 3\)
    10. Domain: \((-\infty, \infty)\); Range: \(y = 3\)

    This page titled 2.1E: Exercises is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Stanislav A. Trunov and Elizabeth J. Hale.

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