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Math Help - Evaluating logs, sin, and piecewise...

  1. #1
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    Evaluating logs, sin, and piecewise...

    Hi, I'm very uncertain of what to do here:

    Let f(x) = log_{2} (x)

    g(x) = sin (\frac{\pi}{2} x)
     h(x) =\left\{\begin{array}{cc}2^{-x},& \mbox{ if } x>0\\-3x^2 - 4x +1, &\mbox{ if }x\leq 0\end{array}\right.

    Evaluate the following:
    (a) (h o f) (4)

    (b) (g/h)(0)


    For the piece-wise defined function, h(x), how can I determine which one to use?
    I tried (h o f)(4), but I'm not certain on it at all:

     <br />
(h o f)(4) = log_{2}(2^{-4})
    <br />
(h o f)(4) = -4 <br />

    Am I anywhere close? Or did I not plug them into each other correctly?
    These confuse me so much.
    Thanks in advanced!
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  2. #2
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    Quote Originally Posted by tiar View Post
    Hi, I'm very uncertain of what to do here:

    Let f(x) = log_{2} (x)

    g(x) = sin (\frac{\pi}{2} x)
     h(x) =\left\{\begin{array}{cc}2^{-x},& \mbox{ if } x>0\\-3x^2 - 4x +1, &\mbox{ if }x\leq 0\end{array}\right.

    Evaluate the following:
    (a) (h o f) (4)

    (b) (g/h)(0)


    For the piece-wise defined function, h(x), how can I determine which one to use?
    I tried (h o f)(4), but I'm not certain on it at all:

     <br />
(h o f)(4) = log_{2}(2^{-4})
    <br />
(h o f)(4) = -4 <br />

    Am I anywhere close? Or did I not plug them into each other correctly?
    These confuse me so much.
    Thanks in advanced!
    1.

    Look at the domain of \log_2{x}. It's x > 0.

    So you've got to use the h(x) where x > 0.


    Therefore

    (h \circ f)(x) = h(f(x))

     = 2^{-\log_2{x}}

     = 2^{\log_2{x^{-1}}}

     = x^{-1}


    (h \circ f)(4) = 4^{-1} = \frac{1}{4}.


    2. \left(\frac{g}{h}\right)(x) = \frac{g(x)}{h(x)}

    \left(\frac{g}{h}\right)(0) = \frac{g(0)}{h(0)}.

    Can you go from here?
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  3. #3
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    Quote Originally Posted by tiar View Post
    Hi, I'm very uncertain of what to do here:

    Let f(x) = log_{2} (x)

    g(x) = sin (\frac{\pi}{2} x)
     h(x) =\left\{\begin{array}{cc}2^{-x},& \mbox{ if } x>0\\-3x^2 - 4x +1, &\mbox{ if }x\leq 0\end{array}\right.

    Evaluate the following:
    (a) (h o f) (4)

    (b) (g/h)(0)


    For the piece-wise defined function, h(x), how can I determine which one to use?
    I tried (h o f)(4), but I'm not certain on it at all:

     <br />
(h o f)(4) = log_{2}(2^{-4})
    <br />
(h o f)(4) = -4 <br />

    Am I anywhere close? Or did I not plug them into each other correctly?
    These confuse me so much.
    Thanks in advanced!
    for (g/h)(0)

    choose the second part of the piece wise function in h(x) as  x\leq 0
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