Evaluating logs, sin, and piecewise...

**tiar** · April 20th 2009, 08:12 PM

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:

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

**Prove It** · April 20th 2009, 08:31 PM

Originally Posted by tiar

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:

$ (h o f)(4) = log_{2}(2^{-4})$
$ (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 $\log_2{x}$ . 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?

**pickslides** · April 20th 2009, 08:45 PM

Originally Posted by tiar

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:

$ (h o f)(4) = log_{2}(2^{-4})$
$ (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 $x\leq 0$

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