I see that you have uploaded an image of a math problem in Indonesian. The problem asks to solve a function division based on two given functions, $f(x) = \sqrt{x} - 3$ and $g(x) = 2x - 8$, followed by four sub-questions (a to d) that deal with the composite functions $h(x) = \frac{f(x)}{g(x)}$ and $p(x) = \frac{f(x)}{g(x)}$, as well as evaluating these functions at specific values of $x$.

Let's solve this step by step:

Given:

• $f(x) = \sqrt{x} - 3$
• $g(x) = 2x - 8$

a. Find $h(x) = \frac{f(x)}{g(x)}$ and the domain (daerah asal) of $h(x)$

• $h(x) = \frac{f(x)}{g(x)} = \frac{\sqrt{x} - 3}{2x - 8}$

The domain of $h(x)$ is determined by two conditions:

1. $f(x) = \sqrt{x} - 3$ implies that $x \geq 0$ (since $\sqrt{x}$ is defined for non-negative values).
2. The denominator $g(x) = 2x - 8$ should not be zero, i.e., $2x - 8 \neq 0 \Rightarrow x \neq 4$.

Thus, the domain of $h(x)$ is $x \geq 0$ but $x \neq 4$. Therefore, the domain is $x \in [0, \infty) \setminus \{4\}$.

b. Find $p(x) = \frac{f(x)}{g(x)}$ and the domain (daerah asal) of $p(x)$

Since both functions $h(x)$ and $p(x)$ are described as the same, this step is essentially the same as part (a):

• $p(x) = \frac{\sqrt{x} - 3}{2x - 8}$
• The domain of $p(x)$ is also $x \in [0, \infty) \setminus \{4\}$.

c. Evaluate $h(4), h(6)$, and $h(8)$

To evaluate $h(x)$ at specific points, plug in the values of $x$ into the function $h(x) = \frac{\sqrt{x} - 3}{2x - 8}$.

• $h(4) = \frac{\sqrt{4} - 3}{2(4) - 8} = \frac{2 - 3}{8 - 8}$ is undefined because the denominator becomes zero. Hence, $h(4)$ is not defined.
• $h(6) = \frac{\sqrt{6} - 3}{2(6) - 8} = \frac{\sqrt{6} - 3}{12 - 8} = \frac{\sqrt{6} - 3}{4}$.
• $h(8) = \frac{\sqrt{8} - 3}{2(8) - 8} = \frac{\sqrt{8} - 3}{16 - 8} = \frac{\sqrt{8} - 3}{8}$.

d. Evaluate $p(4), p(6)$, and $p(8)$

As previously mentioned, $p(x) = h(x)$, so:

• $p(4)$ is also undefined (as $h(4)$ is).
• $p(6) = h(6) = \frac{\sqrt{6} - 3}{4}$.
• $p(8) = h(8) = \frac{\sqrt{8} - 3}{8}$.

Would you like more details on any step, or should we go deeper into any particular part?

5 Related Questions:

1. What would happen if the denominator was a different linear function?
2. Can you find the derivative of $h(x)$ using quotient rule?
3. What are the conditions for $p(x)$ to be continuous over the domain?
4. How do you simplify $\sqrt{8}$ or other square root expressions in function evaluations?
5. How can we find the asymptotes for $h(x)$?

Tip: Always check the denominator for zeroes when determining the domain of rational functions.