If `f(x)` is an odd function, `f(1)=3,f(x+2)=f(x)+f(2),` then the value of `f(3)` is________

Explanation

To solve the problem step by step, we will use the properties of the function given in the question.

### Step 1: Understand the properties of the function
We know that `f(x)` is an odd function. This means that:
\[ f(-x) = -f(x) \]
for all \( x \).

### Step 2: Use the functional equation
We are given the functional equation:
\[ f(x + 2) = f(x) + f(2) \]

### Step 3: Substitute \( x = -1 \) into the functional equation
Let's substitute \( x = -1 \) into the functional equation:
\[ f(-1 + 2) = f(-1) + f(2) \]
This simplifies to:
\[ f(1) = f(-1) + f(2) \]

### Step 4: Use the property of odd functions
Since \( f(x) \) is an odd function, we have:
\[ f(-1) = -f(1) \]
Substituting this into the equation from Step 3 gives:
\[ f(1) = -f(1) + f(2) \]

### Step 5: Solve for \( f(2) \)
Rearranging the equation:
\[ f(1) + f(1) = f(2) \]
\[ 2f(1) = f(2) \]

### Step 6: Substitute the known value of \( f(1) \)
We know from the problem statement that \( f(1) = 3 \):
\[ f(2) = 2 \times 3 = 6 \]

### Step 7: Substitute \( x = 1 \) into the functional equation
Now, let's substitute \( x = 1 \) into the functional equation:
\[ f(1 + 2) = f(1) + f(2) \]
This simplifies to:
\[ f(3) = f(1) + f(2) \]

### Step 8: Substitute the known values
Now we can substitute the values we found:
\[ f(3) = 3 + 6 = 9 \]

### Final Answer
Thus, the value of \( f(3) \) is:
\[ \boxed{9} \]

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