If the function f(x) is defined for all real numbers x as the maximum value of `2x + 4 and 12 + 3x` then for which one of the following values of x will f(x) actually values of x will f(x) actually equal `2x + 4`?

Options

A. `-4`
B. `-5`
C. `-6`
D. `-9`

Answer

To solve the problem, we need to determine the values of \( x \) for which the function \( f(x) \) defined as the maximum of \( 2x + 4 \) and \( 12 + 3x \) is equal to \( 2x + 4 \).

### Step-by-Step Solution:

1. **Define the function**:
\[ f(x) = \max(2x + 4, 12 + 3x) \]

2. **Set up the inequality**:
For \( f(x) \) to equal \( 2x + 4 \), we need:
\[ 2x + 4 \geq 12 + 3x \]

3. **Rearrange the inequality**:
Subtract \( 3x \) from both sides:
\[ 2x - 3x + 4 \geq 12 \]
This simplifies to:
\[ -x + 4 \geq 12 \]

4. **Isolate \( x \)**:
Subtract 4 from both sides:
\[ -x \geq 8 \]
Multiply both sides by -1 (remember to reverse the inequality):
\[ x \leq -8 \]

5. **Conclusion**:
The function \( f(x) \) equals \( 2x + 4 \) when \( x \) is less than or equal to -8. Therefore, we need to find values of \( x \) that satisfy this condition.

6. **Evaluate the options**:
The options given are -4, -5, -6, and -9. We check which of these values are less than -8:
- -4 is greater than -8.
- -5 is greater than -8.
- -6 is greater than -8.
- -9 is less than -8.

Thus, the only suitable option is:
\[ x = -9 \]

### Final Answer:
The value of \( x \) for which \( f(x) = 2x + 4 \) is \( x = -9 \).