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Mathematics LibreTexts

6.4.2: Homework

  • Page ID
    197607
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    Reading Questions

    1. What is a radical function? Give an example of a square root function and a cube root function.
    2. When evaluating radical functions, why is it important to consider the index of the radical (even vs. odd) if the radicand might be negative?
    3. What is the primary condition that the radicand of an even-indexed radical function must satisfy for the function to yield a real number?
    4. What is the domain of an odd-indexed radical function like \(f(x) = \sqrt[3]{x-2}\)?
    5. How do you find the domain of an even-indexed radical function like \(f(x) = \sqrt{3x-4}\)?
    6. If the radicand of an even-indexed radical function is a rational expression, such as \(f(x) = \sqrt{\frac{x+2}{x-1}}\), what conditions must be met to determine the domain?
    7. Describe the general shape of the graph of \(y=\sqrt{x}\) and \(y=\sqrt[3]{x}\).
    8. How do transformations (shifts, stretches, reflections) affect the graph of a basic radical function like \(y=\sqrt{x}\)?
    9. When finding the domain of a function like \(g(x) = \sqrt[4]{2-\sqrt[4]{x+3}}\), which radical's domain requirements should you address first?
    10. What does it mean if, when evaluating a radical function for a specific input, you find the radicand of an even root is negative (e.g., \(f(-2)\) for \(f(x)=\sqrt{2x-1}\))?

    Homework

    In the following exercises, evaluate each function.

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

    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. \(f(x)=\sqrt{4 x-2}\)
    3. \(g(x)=\sqrt{2-3 x}\)
    4. \(g(x)=\sqrt{8-x}\)
    5. \(h(x)=\sqrt{\frac{5}{x-2}}\)
    6. \(h(x)=\sqrt{\frac{6}{x+3}}\)
    7. \(f(x)=\sqrt{\frac{x+3}{x-2}}\)
    8. \(f(x)=\sqrt{\frac{x-1}{x+4}}\)
    9. \(g(x)=\sqrt[3]{8 x-1}\)
    10. \(g(x)=\sqrt[3]{6 x+5}\)
    11. \(f(x)=\sqrt[3]{4 x^{2}-16}\)
    12. \(f(x)=\sqrt[3]{6 x^{2}-25}\)
    13. \(F(x)=\sqrt[4]{8 x+3}\)
    14. \(F(x)=\sqrt[4]{10-7 x}\)
    15. \(G(x)=\sqrt[5]{2 x-1}\)
    16. \(G(x)=\sqrt[5]{6 x-3}\)

    In the following exercises, find the domain of the function, graph the function, and use the graph to determine the range.

    1. \(f(x)=\sqrt{x+1}\)
    2. \(f(x)=\sqrt{x-1}\)
    3. \(g(x)=\sqrt{x+4}\)
    4. \(g(x)=\sqrt{x-4}\)
    5. \(f(x)=\sqrt{x}+2\)
    6. \(f(x)=\sqrt{x}-2\)
    7. \(g(x)=2 \sqrt{x}\)
    8. \(g(x)=3 \sqrt{x}\)
    9. \(f(x)=\sqrt{3-x}\)
    10. \(f(x)=\sqrt{4-x}\)
    11. \(g(x)=-\sqrt{x}\)
    12. \(g(x)=-\sqrt{x}+1\)
    13. \(f(x)=\sqrt[3]{x+1}\)
    14. \(f(x)=\sqrt[3]{x-1}\)
    15. \(g(x)=\sqrt[3]{x+2}\)
    16. \(g(x)=\sqrt[3]{x-2}\)
    17. \(f(x)=\sqrt[3]{x}+3\)
    18. \(f(x)=\sqrt[3]{x}-3\)
    19. \(g(x)=\sqrt[3]{x}\)
    20. \(g(x)=-\sqrt[3]{x}\)
    21. \(f(x)=2 \sqrt[3]{x}\)
    22. \(f(x)=-2 \sqrt[3]{x}\)

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