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Math Help - Symmetric to which axis?

  1. #1
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    Symmetric to which axis?

    The graph of y=cube root of (x^2+1) is symmetric with respect to which of the following?

    I. The X-axis
    II. The Y-axis
    III. The origin

    Now the multiple choice answers are combinations of these numbers, but I do have to show my work. Could someone clear this up without the use of a graphing calculator or a hand drawn graph? I would guess it might have something to do with odd/even functions (which I am not very familiar with). Also, I don't understand how something can be symmetric with respect to the origin. Thanks!
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  2. #2
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    1) Look at it. <== Really. It's an important first step.

    2) Substitute x = -x

    \sqrt[3]{(-x)^{2}+1}\;=\;\sqrt[3]{x^{2}+1}

    3) If that changes nothing, it's symmetric about the Y-Axis. Why is that?

    4) Substitute y = -y

    -y = \sqrt[3]{x^{2}+1} \implies y = -\sqrt[3]{x^{2}+1}

    5) If that DOES change things, then it is NOT symmetric about the X-Axis. Why is that?
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  3. #3
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    Quote Originally Posted by TKHunny View Post
    1) Look at it. <== Really. It's an important first step.

    2) Substitute x = -x

    \sqrt[3]{(-x)^{2}+1}\;=\;\sqrt[3]{x^{2}+1}

    3) If that changes nothing, it's symmetric about the Y-Axis. Why is that?

    4) Substitute y = -y

    -y = \sqrt[3]{x^{2}+1} \implies y = -\sqrt[3]{x^{2}+1}

    5) If that DOES change things, then it is NOT symmetric about the X-Axis. Why is that?
    That's much simpler than I thought it would be. Thanks!
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  4. #4
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    How about the Origin?

    Hint: Look at the first two.
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