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Thread: What type of function is this?

  1. #1
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    What type of function is this?

    y = -x^(1/3)

    What type of function is this? Simplified it is y = i*cube root(x)

    I'm trying to find the symmetry using the formula x = -b/2a.
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  2. #2
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    Quote Originally Posted by mwok View Post
    y = -x^(1/3)

    What type of function is this? Simplified it is y = i*cube root(x)

    I'm trying to find the symmetry using the formula x = -b/2a.
    the function you cite is actually $\displaystyle f(x) = -\sqrt[3]{x}$

    did you mean $\displaystyle y = (-x)^{\frac{1}{3}}$ ?

    if so, this is the same as $\displaystyle f(x) = -\sqrt[3]{x}$

    note that $\displaystyle (-x)^{\frac{1}{3}} = (-1)^{\frac{1}{3}} \cdot x^{\frac{1}{3}} = -1 \cdot x^{\frac{1}{3}}$

    no i is involved ... $\displaystyle (-1)^{\frac{1}{2}} = i$

    $\displaystyle f(x) = -\sqrt[3]{x}$ is an odd function symmetric to the origin.
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    But how did you find the symmetry and odd/even?

    I know that an even/odd power corresponds to a even/odd function, however...the power in this case is a fraction. Without graphing, how can I determined this?

    BTW, what graphing tool did you use to generate that graph?

    Thanks.
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    Quote Originally Posted by mwok View Post
    But how did you find the symmetry and odd/even?

    I know that an even/odd power corresponds to a even/odd function, however...the power in this case is a fraction. Without graphing, how can I determined this?

    BTW, what graphing tool did you use to generate that graph?

    Thanks.
    $\displaystyle f(x)$ is odd if $\displaystyle f(-x) = -f(x)$

    $\displaystyle f(x) = \sqrt[3]{-x} = -\sqrt[3]{x}$

    $\displaystyle f(-x) = \sqrt[3]{-(-x)} = \sqrt[3]{x}$

    $\displaystyle f(-x) = \sqrt[3]{x}$ is the opposite of $\displaystyle f(x) = -\sqrt[3]{x}$ , therefore $\displaystyle f(x)$ is odd.


    free graph program ...

    Graph
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    Quote Originally Posted by mwok View Post
    But how did you find the symmetry and odd/even?
    I know that an even/odd power corresponds to a even/odd function, however...the power in this case is a fraction. Without graphing, how can I determined this?
    The names odd & even most likely came from exponents.
    But exponents really do not determine much.
    For example: $\displaystyle \cos(x)$ is even and $\displaystyle \sin(x)$ is odd.

    Learn the actual definitions: If $\displaystyle f(x)=f(-x)$ then $\displaystyle f$ is even.
    If $\displaystyle -g(x)=g(-x)$ then $\displaystyle g$ is odd.
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  6. #6
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    Thanks, how do you find the symmetry? I cannot use x = -b/2a because that is for polynomials and the function above is not a polynomial.
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    Quote Originally Posted by mwok View Post
    Thanks, how do you find the symmetry?
    First do as I suggested: Learn the basic definitions.
    Then it is simple: odd functions are symmetric about the origin; even functions are symmetric about the y-axis.
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    So, the answer is simply that the symmetry is about the origin?
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  9. #9
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    Quote Originally Posted by mwok View Post
    So, the answer is simply that the symmetry is about the origin?
    Well of course, if it is an odd function.
    $\displaystyle f(x) = - x^{\frac{1}
    {3}} \; \Rightarrow \;f( - x) = - \left( { - x} \right)^{\frac{1}
    {3}} = x^{\frac{1}
    {3}} = - f(x)$ so it is odd.
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