Hi, I just did these two problems and the domain was the same for both problems, am I overlooking something?
h(x) = 4x/Square Root(x^2 - 16)
and
j(x) = Square Root[4x/(x^2 - 16)]
Maybe she's trying to trick me?
The denominator cannot be zero. And the radical must be positive. Thus,
$\displaystyle x^2-16>0$
$\displaystyle x^2>16$
$\displaystyle x>4 \mbox{ or }x<-4$
Over here we need term inside the radical to be positive.j(x) = Square Root[4x/(x^2 - 16)]
First $\displaystyle x^2-16\not = 0$ thus, $\displaystyle x\not = \pm 4$.
And,
$\displaystyle \frac{4x}{x^2-16}>0$
Both positive or both negative
1)$\displaystyle 4x\geq 0 \to x\geq 0$
And,
$\displaystyle x^2-16 >0$ thus, $\displaystyle x>4 \mbox{ or }x<-4$
Combination of those two leads us to,
$\displaystyle x>4$
2)$\displaystyle 4x\leq 0 \to x\leq 0$
$\displaystyle x^2-16 >0$ thus, $\displaystyle x>4 \mbox{ or }x<-4$.
Combination of those two leads us to,
$\displaystyle x<-4$.
Even though they are different functions they have the same solution set.
Typically if you want a polynomial with specific x-intercepts you can simply write them out term by term:
If you have the intercepts 1, 2, 5, and 8 a polynomial that fits this is:
$\displaystyle f(x) = (x - 1)(x - 2)(x - 5)(x - 8)$
Now we want the function $\displaystyle y = \sqrt{k(x)}$ to have the domain $\displaystyle (1, 2) \cup (5, 8)$ which means that this is the part of the (real) x-axis for which k(x) is positive. So let's graph f(x). (See the attachment below.)
Note that in the graph the set $\displaystyle (1, 2) \cup (5, 8)$ is the region where f(x) is negative. So we need to "flip the function over" to make this the region where the function is positive. This is easy. Just define k(x) to be the negative of f(x):
$\displaystyle k(x) = -f(x) = -(x - 1)(x - 2)(x - 5)(x - 8)$.
Then $\displaystyle y = \sqrt{k(x)}$ has the correct domain.
Note: Technically the domain is $\displaystyle [1, 2] \cup [5, 8] $. I couldn't think of a way to define a finite polynomial that excludes the end-points of the sets. (Unless I'm missing the obvious.) You might wish to point this out to your teacher.
-Dan