If `f(x)+2f(1-x)=x^2+2AA x in R,` then `f(x)` given as

Options

A. `((x-1)^(2))/(3)`
B. `((x-2)^(2))/(3)`
C. `x^(2)-1)`
D. `x^(2)-2`

Answer

To solve the equation \( f(x) + 2f(1-x) = x^2 + 2 \) for \( f(x) \), we can follow these steps:

### Step 1: Write down the original equation
We start with the equation:
\[ f(x) + 2f(1-x) = x^2 + 2 \]

### Step 2: Substitute \( x \) with \( 1-x \)
Now, we will substitute \( x \) with \( 1-x \) in the original equation:
\[ f(1-x) + 2f(x) = (1-x)^2 + 2 \]
Calculating \( (1-x)^2 \):
\[ (1-x)^2 = 1 - 2x + x^2 \]
Thus, the equation becomes:
\[ f(1-x) + 2f(x) = 1 - 2x + x^2 + 2 \]
Simplifying this gives:
\[ f(1-x) + 2f(x) = x^2 - 2x + 3 \]

### Step 3: Set up the system of equations
Now we have two equations:
1. \( f(x) + 2f(1-x) = x^2 + 2 \) (Equation 1)
2. \( f(1-x) + 2f(x) = x^2 - 2x + 3 \) (Equation 2)

### Step 4: Multiply Equation 1 by 2
To eliminate \( f(1-x) \), we will multiply Equation 1 by 2:
\[ 2f(x) + 4f(1-x) = 2(x^2 + 2) \]
This simplifies to:
\[ 2f(x) + 4f(1-x) = 2x^2 + 4 \]

### Step 5: Subtract Equation 2 from the modified Equation 1
Now, we subtract Equation 2 from the modified Equation 1:
\[ (2f(x) + 4f(1-x)) - (f(1-x) + 2f(x)) = (2x^2 + 4) - (x^2 - 2x + 3) \]
This simplifies to:
\[ 3f(1-x) = 2x^2 + 4 - x^2 + 2x - 3 \]
Combining like terms gives:
\[ 3f(1-x) = x^2 + 2x + 1 \]
Thus:
\[ f(1-x) = \frac{x^2 + 2x + 1}{3} \]

### Step 6: Substitute back to find \( f(x) \)
Now we can substitute \( f(1-x) \) back into Equation 1:
\[ f(x) + 2\left(\frac{x^2 + 2x + 1}{3}\right) = x^2 + 2 \]
This simplifies to:
\[ f(x) + \frac{2(x^2 + 2x + 1)}{3} = x^2 + 2 \]
Multiplying through by 3 to eliminate the fraction gives:
\[ 3f(x) + 2(x^2 + 2x + 1) = 3(x^2 + 2) \]
This simplifies to:
\[ 3f(x) + 2x^2 + 4x + 2 = 3x^2 + 6 \]
Rearranging gives:
\[ 3f(x) = 3x^2 + 6 - 2x^2 - 4x - 2 \]
Combining like terms results in:
\[ 3f(x) = x^2 - 4x + 4 \]
Thus:
\[ f(x) = \frac{x^2 - 4x + 4}{3} \]

### Step 7: Factor the expression
The expression can be factored:
\[ f(x) = \frac{(x-2)^2}{3} \]

### Final Answer
Thus, the function \( f(x) \) is:
\[ f(x) = \frac{(x-2)^2}{3} \]