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Thread: 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:

    $\displaystyle Let f(x) = log_{2} (x)$

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

    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:

    $\displaystyle
    (h o f)(4) = log_{2}(2^{-4})$
    $\displaystyle
    (h o f)(4) = -4
    $

    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:

    $\displaystyle Let f(x) = log_{2} (x)$

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

    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:

    $\displaystyle
    (h o f)(4) = log_{2}(2^{-4})$
    $\displaystyle
    (h o f)(4) = -4
    $

    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 $\displaystyle \log_2{x}$. It's $\displaystyle x > 0$.

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


    Therefore

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

    $\displaystyle = 2^{-\log_2{x}}$

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

    $\displaystyle = x^{-1}$


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


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

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

    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:

    $\displaystyle Let f(x) = log_{2} (x)$

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

    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:

    $\displaystyle
    (h o f)(4) = log_{2}(2^{-4})$
    $\displaystyle
    (h o f)(4) = -4
    $

    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 $\displaystyle x\leq 0 $
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