If `f(x)=2x+1`, and `f(x+2)+f(x)=x`, the value of x is

Options

A. `-2`
B. `-1`
C. `-(1)/(2)`
D. `(1)/(2)`

Explanation

To solve the equation given in the problem, we will follow these steps:

1. **Identify the function**: We are given that \( f(x) = 2x + 1 \).

2. **Find \( f(x + 2) \)**: To find \( f(x + 2) \), we substitute \( x + 2 \) into the function:
\[ f(x + 2) = 2(x + 2) + 1 = 2x + 4 + 1 = 2x + 5 \]

3. **Set up the equation**: According to the problem, we need to find \( f(x + 2) + f(x) \):
\[ f(x + 2) + f(x) = (2x + 5) + (2x + 1) = 2x + 5 + 2x + 1 = 4x + 6 \]

4. **Equate to x**: We know from the problem statement that \( f(x + 2) + f(x) = x \). Therefore, we set up the equation:
\[ 4x + 6 = x \]

5. **Rearrange the equation**: To isolate \( x \), we can subtract \( x \) from both sides:
\[ 4x + 6 - x = 0 \implies 3x + 6 = 0 \]

6. **Solve for x**: Now, we will solve for \( x \) by isolating it:
\[ 3x = -6 \implies x = -2 \]

Thus, the value of \( x \) is \( -2 \).