To find the domains of the functions:

1. Domain of $f(x) = \sqrt{8 - x}$

• The square root function is only defined for non-negative values inside the square root. Therefore, we need: $8 - x \geq 0$ Solving this inequality: $x \leq 8$ So, the domain of $f(x)$ is: $(-\infty, 8]$

2. Domain of $g(x) = \frac{1}{x - 3}$

• The function $g(x)$ is a rational function, and it is undefined when the denominator equals zero. So, we need to exclude the value where $x - 3 = 0$, which is $x = 3$. The domain of $g(x)$ is all real numbers except 3: $(-\infty, 3) \cup (3, \infty)$

Final Results:

• Domain of $f(x)$: $(-\infty, 8]$
• Domain of $g(x)$: $(-\infty, 3) \cup (3, \infty)$

Would you like any details or have any further questions?

Here are 5 related questions to explore:

1. What is the range of $f(x) = \sqrt{8 - x}$?
2. Can you find the composition of $f(g(x))$?
3. How do we determine the domain of a product of two functions?
4. What happens to the graph of $f(x) = \sqrt{8 - x}$ as $x$ approaches 8?
5. What is the vertical asymptote of $g(x) = \frac{1}{x - 3}$?

Tip: Always check for restrictions like square roots or denominators when finding a function's domain.