Part C: What is the domain of f divided by g of x? (2 points)

To determine the domain of \( \frac{f}{g}(x) \), we first need to define the functions \( f(x) \) and \( g(x) \) and then establish where the function \( \frac{f(x)}{g(x)} \) is defined.

Given:
\[ f(x) = 4x + 9 \]
\[ g(x) = x + 2 \]

The function \( \frac{f}{g}(x) \) is defined as:
\[ \frac{f}{g}(x) = \frac{4x + 9}{x + 2} \]

The domain of this function is all real numbers except where the denominator is zero because division by zero is undefined. Therefore, we need to find the values of \( x \) for which \( g(x) = x + 2 \) equals zero.

Set the denominator equal to zero and solve for \( x \):
\[ x + 2 = 0 \]
\[ x = -2 \]

Thus, \( \frac{f}{g}(x) \) is undefined at \( x = -2 \).

### Domain of \( \frac{f}{g}(x) \)

The domain of \( \frac{f}{g}(x) \) is all real numbers except \( x = -2 \). In interval notation, this is expressed as:

\[ \text{Domain} = (-\infty, -2) \cup (-2, \infty) \]

### Explanation

In terms of the scenario, the function \( f(x) = 4x + 9 \) represents the number of animals adopted at one shelter, and \( g(x) = x + 2 \) represents the number of animals adopted at another shelter over \( x \) months. The combined function \( \frac{f}{g}(x) \) would represent the ratio of the number of animals adopted at the first shelter to the number of animals adopted at the second shelter. However, this ratio is undefined when \( x = -2 \) because it would imply the second shelter adopted zero animals, which is not possible in this context.

Do you want more details or have any questions?

Here are some additional questions to expand on this information:

1. What does the numerator \( f(x) \) and denominator \( g(x) \) represent in the context of the shelters?
2. How does the value of \( \frac{f}{g}(x) \) change as \( x \) increases?
3. What happens to the value of \( \frac{f}{g}(x) \) as \( x \) approaches \(-2\) from the left and right?
4. How would you interpret the value of \( \frac{f}{g}(x) \) for a specific month \( x \)?
5. How does the exclusion of \( x = -2 \) affect the interpretation of the combined function in the scenario?
6. What does the slope of \( \frac{f}{g}(x) \) indicate about the adoption rates of the two shelters?
7. Can you identify any asymptotes in the function \( \frac{f}{g}(x) \) and explain their significance?
8. How would changes in the adoption rates at either shelter affect the domain and range of \( \frac{f}{g}(x) \)?

**Tip:** When determining the domain of a function involving a ratio, always check for values that make the denominator zero, as these are the points where the function is undefined.

Learn how to determine the domain of the function f divided by g of x with a step-by-step explanation. This problem involves functions, rational functions, and understanding domain restrictions. Suitable for Grades 10-12.

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