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5.3E: Exercises

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    Practice Makes Perfect

    Exercise \(\PageIndex{17}\) Evaluate a Radical Function

    In the following exercises, evaluate each function.

    1. \(f(x)=\sqrt{4 x-4}\), find
      1. \(f(5)\)
      2. \(f(0)\)
    2. \(g(x)=\sqrt{6 x+1}\), find
      1. \(g(4)\)
      2. \(g(8)\)
    3. \(F(x)=\sqrt{3-2 x}\), find
      1. \(F(1)\)
      2. \(F(-11)\)
    4. \(G(x)=\sqrt{5 x-1}\), find
      1. \(G(5)\)
      2. \(G(2)\)
    5. \(g(x)=\sqrt[3]{2 x-4}\), find
      1. \(g(6)\)
      2. \(g(-2)\)
    6. \(h(x)=\sqrt[3]{x^{2}-4}\), find
      1. \(h(-2)\)
      2. \(h(6)\)
    7. For the function \(f(x)=\sqrt[4]{2 x^{3}}\), find
      1. \(f(0)\)
      2. \(f(2)\)
    8. For the function \(g(x)=\sqrt[4]{4-4 x}\), find
      1. \(g(1)\)
      2. \(g(-3)\)
    Answer

    1.

    1. \(f(5)=4\)
    2. \(f(5)=2i\)

    2.

    1. \(g(4)=5\)
    2. \(g(8)=7\)

    3.

    1. \(F(1)=1\)
    2. \(F(-11)=5\)

    4.

    1. \(G(5)=2 \sqrt{6}\)
    2. \(G(2)=3\)

    5.

    1. \(g(6)=2\)
    2. \(g(-2)=-2\)

    6.

    1. \(h(-2)=0\)
    2. \(h(6)=2 \sqrt[3]{4}\)

    7.

    1. \(f(0)=0\)
    2. \(f(2)=2\)

    8.

    1. \(g(1)=0\)
    2. \(g(-3)=2\)
    Exercise \(\PageIndex{18}\) Find the Domain of a Radical Function

    In the following exercises, find the domain of the function and write the domain in interval notation.

    1. \(f(x)=\sqrt{3 x-1}\)
    2. \(g(x)=\sqrt{2-3 x}\)
    3. \(h(x)=\sqrt{x-2}\)
    4. \(f(x)=\sqrt{-x+3}\)
    5. \(g(x)=\sqrt[3]{8 x-1}\)
    6. \(f(x)=\sqrt[3]{4 x^{2}-16}\)
    7. \(F(x)=\sqrt[4]{8 x+3}\)
    8. \(G(x)=\sqrt[5]{2 x-1}\)
    Answer
    1. \(\left[\frac{1}{3}, \infty\right)\)
    2. \(\left(-\infty, \frac{2}{3}\right]\)
    3. \([2, \infty)\)
    4. \((-\infty,3]\)
    5. \((-\infty, \infty)\)
    6. \((-\infty, \infty)\)
    7. \(\left[-\frac{3}{8}, \infty\right)\)
    8. \((-\infty, \infty)\)
    Exercise \(\PageIndex{19}\) graph radical functions

    In the following exercises,

    1. find the domain of the function
    2. graph the function
    3. use the graph to determine the range
      1. \(f(x)=\sqrt{x+1}\)
      2. \(g(x)=\sqrt{x+4}\)
      3. \(f(x)=\sqrt{x}+2\)
      4. \(g(x)=2 \sqrt{x}\)
      5. \(f(x)=\sqrt{3-x}\)
      6. \(g(x)=-\sqrt{x}\)
      7. \(f(x)=\sqrt[3]{x+1}\)
      8. \(g(x)=\sqrt[3]{x+2}\)
      9. \(f(x)=\sqrt[3]{x}+3\)
      10. \(g(x)=\sqrt[3]{x}\)
      11. \(f(x)=2 \sqrt[3]{x}\)
    Answer

    17.

    1. domain: \([-1, \infty)\)

    2. The figure shows a square root function graph on the x y-coordinate plane. The x-axis of the plane runs from negative 1 to 7. The y-axis runs from negative 2 to 10. The function has a starting point at (negative 1, 0) and goes through the points (0, 1) and (3, 2).
      Figure 8.7.8
    3. \([0, \infty)\)

    18.

    1. domain: \([-4, \infty)\)

    2. The figure shows a square root function graph on the x y-coordinate plane. The x-axis of the plane runs from negative 4 to 4. The y-axis runs from negative 2 to 6. The function has a starting point at (negative 4, 0) and goes through the points (negative 3, 1) and (0, 2).
      Figure 8.7.9
    3. \([0, \infty)\)

    19.

    1. domain: \([0, \infty)\)

    2. The figure shows a square root function graph on the x y-coordinate plane. The x-axis of the plane runs from 0 to 8. The y-axis runs from 0 to 8. The function has a starting point at (0, 2) and goes through the points (1, 3) and (4, 4).
      Figure 8.7.10
    3. \([2, \infty)\)

    20.

    1. domain: \([0, \infty)\)

    2. The figure shows a square root function graph on the x y-coordinate plane. The x-axis of the plane runs from 0 to 8. The y-axis runs from 0 to 8. The function has a starting point at (0, 0) and goes through the points (1, 2) and (4, 4).
      Figure 8.7.11
    3. \([0, \infty)\)

    21.

    1. domain: \((-\infty, 3]\)

    2. The figure shows a square root function graph on the x y-coordinate plane. The x-axis of the plane runs from negative 6 to 4. The y-axis runs from 0 to 8. The function has a starting point at (3, 0) and goes through the points (2, 1), (negative 1, 2), and (negative 6, 3).
      Figure 8.7.12
    3. \([0, \infty)\)

    22.

    1. domain: \([0, \infty)\)

    2. The figure shows a square root function graph on the x y-coordinate plane. The x-axis of the plane runs from 0 to 8. The y-axis runs from negative 8 to 0. The function has a starting point at (0, 0) and goes through the points (1, negative 1) and (4, negative 2).
      Figure 8.7.13
    3. \((-\infty, 0]\)

    23.

    1. domain: \((-\infty, \infty)\)

    2. The figure shows a cube root function graph on the x y-coordinate plane. The x-axis of the plane runs from negative 4 to 4. The y-axis runs from negative 4 to 4. The function has a center point at (negative 1, 0) and goes through the points (negative 2, negative 1) and (0, 1).
      Figure 8.7.14
    3. \((-\infty, \infty)\)

    24.

    1. domain: \((-\infty, \infty)\)

    2. The figure shows a cube root function graph on the x y-coordinate plane. The x-axis of the plane runs from negative 4 to 4. The y-axis runs from negative 4 to 4. The function has a center point at (negative 4, 0) and goes through the points (negative 3, negative 1) and (negative 1, 1).
      Figure 8.7.15
    3. \((-\infty, \infty)\)

    25.

    1. domain: \((-\infty, \infty)\)

    2. The figure shows a cube root function graph on the x y-coordinate plane. The x-axis of the plane runs from negative 4 to 4. The y-axis runs from negative 2 to 6. The function has a center point at (0, 3) and goes through the points (negative 1, 2) and (1, 4).
      Figure 8.7.16
    3. \((-\infty, \infty)\)

    26.

    1. domain: \((-\infty, \infty)\)

    2. The figure shows a cube root function graph on the x y-coordinate plane. The x-axis of the plane runs from negative 4 to 4. The y-axis runs from negative 4 to 4. The function has a center point at (0, 0) and goes through the points (1, 1) and (negative 1, negative 1).
      Figure 8.7.17
    3. \((-\infty, \infty)\)

    27.

    1. domain: \((-\infty, \infty)\)

    2. The figure shows a cube root function graph on the x y-coordinate plane. The x-axis of the plane runs from negative 4 to 4. The y-axis runs from negative 4 to 4. The function has a center point at (0, 0) and goes through the points (1, 2) and (negative 1, negative 2).
      Figure 8.7.18
    3. \((-\infty, \infty)\)
    Exercise \(\PageIndex{20}\) writing exercises
    1. Explain how to find the domain of a fourth root function.
    2. Explain how to find the domain of a fifth root function.
    3. Explain why \(y=\sqrt[3]{x}\) is a function.
    4. Explain why the process of finding the domain of a radical function with an even index is different from the process when the index is odd.
    Answer

    Answers may vary

    This page titled 5.3E: 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 via source content that was edited to the style and standards of the LibreTexts platform.