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Math Help - relations problem

  1. #1
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    Unhappy relations problem



    so this is one of those problems I absolutely did not solve after numerous attempts. The relations are already troubling me.. yet they also want me to find the domain and range.. I'm lost =[
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  2. #2
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    a)

    For f(x) \cdot g(x), you simply multiply.

    (f \cdot g)(x) = { \left(\dfrac {6 \sqrt{x}+x^{2} }{\sqrt[5]{x-2}}\right)\left(\dfrac{3}{5\sqrt[3]{x^{2}}}\right)} = {\dfrac{3 \left( 6\sqrt{x}+x^{2}\right)}{\left( 5\sqrt[3]{x^{2}}\right) \left( \sqrt[5]{x-2}\right)}} = {\dfrac{3x^{\frac{1}{2}} \left(x^{\frac{3}{2}} + 6\right)}{5x^{\frac{2}{3}}\left(x-2\right)^{\frac{1}{5}}}} =
    {\dfrac{3\left(\sqrt[3]{x^{2}}+6\right)}{5 \sqrt[6]{x}\sqrt[5]{x-2}}}.

    To find the domain, notice that if the denominator equals 0, it makes the function undefined. So 5\sqrt[6]{x}\sqrt[5]{x-2}=0, x = 0 or x = 2. So thus far, the domain is x \neq 0 and x \neq 2. But also notice the square root in the denominator, so x cannot be negative. So the domain is x > 0 and x \neq 2.

    The range is a bit trickier. I'm not really sure how to get it either.
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  3. #3
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    After you did the 2nd step, I don't know how you got 3x^1/2(x^3/2 +6). I also can't figure out what you did after that ^.^''

    Thanks
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  4. #4
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    Remember the properties of exponents, x^{\frac{a}{b}} = \sqrt[b]{x^{a}}, and the distributive property.

    From 3\left(6\sqrt{x}+x^{2}\right).

    Convert the radical to an exponent,
    3\left(6x^{\frac{1}{2}}+x^{2}\right).

    Factor out the x^{\frac{1}{2}},
    3x^{\frac{1}{2}}\left(x^{\frac{3}{2}} + 6\right)
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  5. #5
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    Thank you so much! Yay I have 1/4 of this problem solved haha. Does anyone have any ideas on how to solve the other parts of this problem? (especially c and d)
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  6. #6
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    C and D are compositions of functions. You might want to check here for reference, Composition of Functions

    For C)

    (f \circ g)(x)={f(g(x))} = {\dfrac{6 \sqrt{\dfrac{3}{5\sqrt[3]{x^{2}}}} + \left( \dfrac{3}{5 \sqrt[3]{x^{2}}} \right)^{2}} {\sqrt[5]{5\sqrt[3]{x^{2}}-2}}}

    This gets really really messy; I didn't even bother trying to work this out by hand; but according to my TI-89 calculator, the composition should look something like this:

    \dfrac{3 \sqrt[5]{5} \left( 3 \sqrt[3]{|x|} + 10 \sqrt{15} \sqrt[3]{x^{4}} \right)} {25 \sqrt[5]{x^{6}} \sqrt[5]{10\sqrt[3]{x^{2}}-3}\sqrt[3]{|x|}}

    This is obviously very messy to work with. I would suggest using a graphing calculator to approximate the domain and range.

    Good Luck.
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  7. #7
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    Wow.. that does look very messy O_O .. I'll just go as far as i can.. haha

    Thanks for the help!
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  8. #8
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    But the concept for C and D is really simple. It's just those functions are really messy to begin with.

    Say f(x) = x+2 and g(x)= x^{2}

    So
    (f \circ g)(x) = f(g(x)) = f(x^{2}) = x^2 + 2.

    The domain and range of this is trivial.
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  9. #9
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    I'm pretty familiar with that concept =] This problem is just the toughest i've seen.
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