### 1. Break Down the Problem
To find the domain of the composite function $ f \circ g $, we need to:
1. Determine the domain of the function $ g(x) = x + 3 $.
2. Determine the function $ f(x) $.
3. Find the composite function $ f(g(x)) $.
4. Identify any restrictions on the domain of $ f(g(x)) $.

### 2. Relevant Concepts
1. The domain of a function is the set of all possible input values (x-values) that will not cause any undefined behavior, such as division by zero or taking the square root of a negative number.
2. For the composition of functions $ f \circ g $, the domain is determined by the domain of $ g $ and any restrictions that $ f $ imposes when $ g(x) $ is substituted into $ f $.

### 3. Analysis and Detail
1. **Determine the Domain of $ g(x) $**:
$$
g(x) = x + 3
$$
The function $ g(x) $ is a linear function, so its domain is all real numbers:
$$
\text{Domain of } g = (-\infty, \infty)
$$

2. **Consider the Function $ f(x) $**:
Since $ f(x) $ is not provided in the original problem, let's denote $ f(x) $ as a general function. We need to analyze the effect of substituting $ g(x) $ into $ f(x) $.

3. **Substituting $ g(x) $ into $ f(x) $**:
$$
f(g(x)) = f(x + 3)
$$

Without knowing the specific expression for $ f(x) $, we cannot solve this exactly. Therefore, we typically must know additional restrictions from $ f(x) $ to find the domain of $ f \circ g $.

4. **Based on Possible Answer Choices**: We are given multiple choice answers, which imply certain restrictions. Therefore, we need to analyze the choices to determine the valid domain based on typical function constraints.

### 4. Verify and Summarize
- Since the specific form of $ f(x) $ is missing, we cannot complete this part definitively. However, we can analyze the answer choices based on common domain restrictions with linear functions.

**Possible Domains**:
1. $(-∞, -6]$ ∪ $[-6, ∞)$ indicates restriction at $ x = -6 $.
2. $(-∞, -5) ∪ (-5, ∞)$ indicates restriction at $ x = -5 $.
3. $(-∞, ∞)$ indicates no restriction.
4. $(-∞, -8) ∪ (-8, ∞)$ indicates restriction at $ x = -8 $.

Assuming $ f(x) $ might have a vertical asymptote or undefined behavior at certain values:
- If $ f $ were to restrict at a value like -5 or -6 or -8, we'd select those options as possible domain results.

### Final Answer
Since we lacked information on $ f(x) $, we cannot definitively state which choice is correct. However, based on $ g(x) = x+3 $ being unrestricted, the most reasonable assumption for a function that might restrict the domain would likely be:

$$
\text{The answer is most likely } (-\infty, -5) \cup (-5, \infty) \text{ if we consider standard asymptotic behavior.}
$$

Question

### 1. Break Down the Problem
To find the domain of the composite function $ f \circ g $, we need to:
1. Determine the domain of the function $ g(x) = x + 3 $.
2. Determine the function $ f(x) $.
3. Find the composite function $ f(g(x)) $.
4. Identify any restrictions on the domain of $ f(g(x)) $.

### 2. Relevant Concepts
1. The domain of a function is the set of all possible input values (x-values) that will not cause any undefined behavior, such as division by zero or taking the square root of a negative number.
2. For the composition of functions $ f \circ g $, the domain is determined by the domain of $ g $ and any restrictions that $ f $ imposes when $ g(x) $ is substituted into $ f $.

### 3. Analysis and Detail
1. **Determine the Domain of $ g(x) $**:
   $$
   g(x) = x + 3 
   $$
   The function $ g(x) $ is a linear function, so its domain is all real numbers:
   $$
   \text{Domain of } g = (-\infty, \infty)
   $$

2. **Consider the Function $ f(x) $**:
   Since $ f(x) $ is not provided in the original problem, let's denote $ f(x) $ as a general function. We need to analyze the effect of substituting $ g(x) $ into $ f(x) $.

3. **Substituting $ g(x) $ into $ f(x) $**:
   $$
   f(g(x)) = f(x + 3)
   $$

Without knowing the specific expression for $ f(x) $, we cannot solve this exactly. Therefore, we typically must know additional restrictions from $ f(x) $ to find the domain of $ f \circ g $.

4. **Based on Possible Answer Choices**: We are given multiple choice answers, which imply certain restrictions. Therefore, we need to analyze the choices to determine the valid domain based on typical function constraints.

### 4. Verify and Summarize
- Since the specific form of $ f(x) $ is missing, we cannot complete this part definitively. However, we can analyze the answer choices based on common domain restrictions with linear functions.

**Possible Domains**:
1. $(-∞, -6]$ ∪ $[-6, ∞)$ indicates restriction at $ x = -6 $.
2. $(-∞, -5) ∪ (-5, ∞)$ indicates restriction at $ x = -5 $.
3. $(-∞, ∞)$ indicates no restriction.
4. $(-∞, -8) ∪ (-8, ∞)$ indicates restriction at $ x = -8 $.

Assuming $ f(x) $ might have a vertical asymptote or undefined behavior at certain values:
- If $ f $ were to restrict at a value like -5 or -6 or -8, we'd select those options as possible domain results.

### Final Answer
Since we lacked information on $ f(x) $, we cannot definitively state which choice is correct. However, based on $ g(x) = x+3 $ being unrestricted, the most reasonable assumption for a function that might restrict the domain would likely be:

$$
\text{The answer is most likely } (-\infty, -5) \cup (-5, \infty) \text{ if we consider standard asymptotic behavior.}
$$

Knowee AI · Accepted Answer