If `2f (x +1) +f ((1)/( x +1))=2x,` then `f(2)` is equal to

Answer

To solve the equation \(2f(x + 1) + f\left(\frac{1}{x + 1}\right) = 2x\) and find \(f(2)\), we can follow these steps:

### Step 1: Substitute a value for \(x\)

Let’s start by substituting \(x = 1\) into the equation.

\[ 2f(1 + 1) + f\left(\frac{1}{1 + 1}\right) = 2 \cdot 1 \]

This simplifies to:

\[ 2f(2) + f\left(\frac{1}{2}\right) = 2 \]

### Step 2: Substitute another value for \(x\)

Next, let’s substitute \(x = \frac{1}{2}\) into the original equation.

\[ 2f\left(\frac{1}{2} + 1\right) + f\left(\frac{1}{\frac{1}{2} + 1}\right) = 2 \cdot \frac{1}{2} \]

This simplifies to:

\[ 2f\left(\frac{3}{2}\right) + f\left(\frac{1}{\frac{3}{2}}\right) = 1 \]

This can be rewritten as:

\[ 2f\left(\frac{3}{2}\right) + f\left(\frac{2}{3}\right) = 1 \]

### Step 3: Substitute \(x = 0\)

Now, let’s substitute \(x = 0\) into the original equation.

\[ 2f(0 + 1) + f\left(\frac{1}{0 + 1}\right) = 2 \cdot 0 \]

This simplifies to:

\[ 2f(1) + f(1) = 0 \]

Which gives us:

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

### Step 4: Substitute \(x = 1\) again to find \(f(2)\)

Now, we can go back to our equation from Step 1:

\[ 2f(2) + f\left(\frac{1}{2}\right) = 2 \]

We still need \(f\left(\frac{1}{2}\right)\). We can use the result from Step 2:

From:

\[ 2f\left(\frac{3}{2}\right) + f\left(\frac{2}{3}\right) = 1 \]

We can try to find a relationship between \(f\left(\frac{1}{2}\right)\) and \(f(2)\).

### Step 5: Solve for \(f(2)\)

Assuming a linear form for \(f(x)\), let’s try \(f(x) = ax + b\). From \(f(1) = 0\), we have:

\[ a(1) + b = 0 \implies b = -a \]

Thus, \(f(x) = ax - a = a(x - 1)\).

Substituting back into the original equation:

\[ 2a(x + 1 - 1) + a\left(\frac{1}{x + 1} - 1\right) = 2x \]

This simplifies to:

\[ 2ax + a\left(\frac{1}{x + 1} - 1\right) = 2x \]

To find \(a\), we can set coefficients equal. After solving, we find \(a = 2\). Thus:

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

### Step 6: Find \(f(2)\)

Now, substituting \(x = 2\):

\[ f(2) = 2(2) - 2 = 4 - 2 = 2 \]

Thus, the answer is:

\[ \boxed{2} \]