The maximum value of
f(x) , if f(x) + `f ((1)/(x)) = (1)/(x) , x in ` domain of f

Options

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

Answer

To solve the problem, we need to find the maximum value of the function \( f(x) \) given the equation:

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

### Step 1: Substitute \( x \) with \( \frac{1}{x} \)

We start by substituting \( x \) with \( \frac{1}{x} \) in the original equation:

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

### Step 2: Set up the two equations

Now we have two equations:
1. \( f(x) + f\left(\frac{1}{x}\right) = \frac{1}{x} \) (Equation 1)
2. \( f\left(\frac{1}{x}\right) + f(x) = x \) (Equation 2)

### Step 3: Analyze the equations

From both equations, we can rearrange them:
- From Equation 1: \( f\left(\frac{1}{x}\right) = \frac{1}{x} - f(x) \)
- From Equation 2: \( f(x) = x - f\left(\frac{1}{x}\right) \)

### Step 4: Substitute \( f\left(\frac{1}{x}\right) \) into Equation 1

Substituting \( f\left(\frac{1}{x}\right) \) from Equation 1 into Equation 2 gives:

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

This simplifies to:

\[ \frac{1}{x} = x \]

### Step 5: Solve for \( x \)

Multiplying both sides by \( x \) (assuming \( x \neq 0 \)) gives:

\[ 1 = x^2 \]

Thus, we find:

\[ x = \pm 1 \]

### Step 6: Determine the values of \( f(x) \)

Now we will evaluate \( f(x) \) at these points.

1. For \( x = 1 \):
\[ f(1) + f(1) = 1 \implies 2f(1) = 1 \implies f(1) = \frac{1}{2} \]

2. For \( x = -1 \):
\[ f(-1) + f(-1) = -1 \implies 2f(-1) = -1 \implies f(-1) = -\frac{1}{2} \]

### Step 7: Find the maximum value

Now we compare the values obtained:
- \( f(1) = \frac{1}{2} \)
- \( f(-1) = -\frac{1}{2} \)

The maximum value of \( f(x) \) is:

\[ \text{Maximum value} = \frac{1}{2} \]

### Final Answer

Thus, the maximum value of \( f(x) \) is:

\[ \boxed{\frac{1}{2}} \]