The function `f (x)` satisfy the equation `f (1-x)+ 2f (x) =3x AA x in R,` then `f (0)=`

Options

A. `-2`
B. `-1`
C. 0
D. 1

Answer

To solve the problem, we need to find the value of \( f(0) \) given the functional equation:

\[ f(1-x) + 2f(x) = 3x \quad \text{for all } x \in \mathbb{R} \]

### Step 1: Substitute \( x = 1 - x \)

Let's first substitute \( x \) with \( 1 - x \) in the original equation:

\[ f(1 - (1 - x)) + 2f(1 - x) = 3(1 - x) \]

This simplifies to:

\[ f(x) + 2f(1 - x) = 3 - 3x \]

Now we have two equations:

1. \( f(1 - x) + 2f(x) = 3x \) (Equation 1)
2. \( f(x) + 2f(1 - x) = 3 - 3x \) (Equation 2)

### Step 2: Rearranging the equations

From Equation 1, we can express \( f(1 - x) \):

\[ f(1 - x) = 3x - 2f(x) \]

Now, substitute this expression for \( f(1 - x) \) into Equation 2:

\[ f(x) + 2(3x - 2f(x)) = 3 - 3x \]

### Step 3: Simplifying the equation

Expanding the equation gives:

\[ f(x) + 6x - 4f(x) = 3 - 3x \]

Combining like terms:

\[ -3f(x) + 6x = 3 - 3x \]

### Step 4: Isolating \( f(x) \)

Now, rearranging the equation to isolate \( f(x) \):

\[ -3f(x) = 3 - 3x - 6x \]

This simplifies to:

\[ -3f(x) = 3 - 9x \]

Dividing both sides by -3:

\[ f(x) = 3x - 1 \]

### Step 5: Finding \( f(0) \)

Now that we have the function \( f(x) \), we can find \( f(0) \):

\[ f(0) = 3(0) - 1 = -1 \]

Thus, the value of \( f(0) \) is:

\[ \boxed{-1} \]