Let f be a function with domain [-3, 5] and let g (x) = `| 3x + 4 |`, Then the domain of (fog) (x) is

Options

A. `(-3, (1)/(3))`
B. `[-3, (1)/(3)]`
C. `[-3, (1)/(3))`
D. None of these

Answer

To find the domain of the composite function \( (f \circ g)(x) \), we need to ensure that the output of \( g(x) \) falls within the domain of \( f \).

1. **Identify the domain of \( f \)**:
The function \( f \) has a domain of \([-3, 5]\).

2. **Define the function \( g(x) \)**:
The function \( g(x) = |3x + 4| \).

3. **Determine the range of \( g(x) \)**:
Since \( g(x) \) is the absolute value function, it will always produce non-negative outputs. Therefore, the minimum value of \( g(x) \) occurs when \( 3x + 4 = 0 \):
\[ 3x + 4 = 0 \implies 3x = -4 \implies x = -\frac{4}{3} \]
At \( x = -\frac{4}{3} \), \( g\left(-\frac{4}{3}\right) = 0 \).

4. **Find the maximum value of \( g(x) \)**:
As \( x \) approaches positive infinity, \( g(x) \) will also approach positive infinity. Thus, the range of \( g(x) \) is \([0, \infty)\).

5. **Set the range of \( g(x) \) within the domain of \( f \)**:
We need to find the values of \( x \) such that \( g(x) \) falls within the domain of \( f \):
\[ 0 \leq g(x) \leq 5 \]
This translates to finding when \( g(x) \leq 5 \):
\[ |3x + 4| \leq 5 \]

6. **Solve the inequality**:
This absolute value inequality can be split into two cases:
- Case 1: \( 3x + 4 \leq 5 \)
- Case 2: \( 3x + 4 \geq -5 \)

**For Case 1**:
\[ 3x + 4 \leq 5 \implies 3x \leq 1 \implies x \leq \frac{1}{3} \]

**For Case 2**:
\[ 3x + 4 \geq -5 \implies 3x \geq -9 \implies x \geq -3 \]

7. **Combine the results**:
From the two cases, we have:
\[ -3 \leq x \leq \frac{1}{3} \]

8. **Conclusion**:
Thus, the domain of \( (f \circ g)(x) \) is \([-3, \frac{1}{3}]\).