# Thread: Evaluating logs, sin, and piecewise...

1. ## 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.

2. Originally Posted by tiar
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.
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?

3. Originally Posted by tiar
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.
choose the second part of the piece wise function in h(x) as $\displaystyle x\leq 0$