The image describes two functions $f(x)$ and $g(x)$, and it asks to find the quotient $\frac{f}{g}(x)$, simplify it, and determine its domain.

Given Functions:

• $f(x) = \frac{x - 6}{x + 4}$
• $g(x) = \frac{x}{x + 4}$

Step 1: Find $\frac{f}{g}(x)$

We compute $\frac{f(x)}{g(x)}$ as follows: $\frac{f(x)}{g(x)} = \frac{\frac{x - 6}{x + 4}}{\frac{x}{x + 4}} = \frac{x - 6}{x}$

Thus, the simplified form of $\frac{f}{g}(x)$ is: $\frac{f}{g}(x) = \frac{x - 6}{x}$

Step 2: Domain of $\frac{f}{g}(x)$

The domain of a rational function is determined by the values of $x$ that do not make the denominator zero. We must exclude any $x$ that makes the original denominators in both $f(x)$ and $g(x)$ equal to zero.

1. In $f(x) = \frac{x - 6}{x + 4}$, the denominator $x + 4 = 0$, so $x = -4$ is excluded.
2. In $g(x) = \frac{x}{x + 4}$, the denominator $x + 4 = 0$, so again, $x = -4$ is excluded.
3. For $\frac{f}{g}(x) = \frac{x - 6}{x}$, the denominator $x = 0$ must be excluded as well.

Thus, the domain excludes both $x = -4$ and $x = 0$.

Final Domain:

The domain is all real numbers except $x = -4$ and $x = 0$. In interval notation, this is: $(-\infty, -4) \cup (-4, 0) \cup (0, \infty)$

Final Answer:

$\frac{f}{g}(x) = \frac{x - 6}{x}$ Domain: $(-\infty, -4) \cup (-4, 0) \cup (0, \infty)$

Would you like more details or have any questions?

Here are 5 related questions to explore:

1. How do you handle division by zero when finding the domain of rational functions?
2. What happens to the domain if the numerator has no variables?
3. Can you give an example of a rational function with no restrictions on its domain?
4. How do you find the domain of a composite function?
5. How would the domain change if $g(x)$ had a different denominator?

Tip: Always remember to check for restrictions from both the numerator and denominator when finding the domain of rational functions.