If `f(x)` is a continuous function such that its value `AA x in R` is a rational number and `f(1)+f(2)=6`, then the value of `f(3)` is equal to

Options

A. 3
B. 9
C. 2
D. 4

Answer

To solve the problem, we need to analyze the properties of the function \( f(x) \) given the conditions stated in the question.

### Step-by-Step Solution:

1. **Understand the Function**:
We know that \( f(x) \) is a continuous function that takes rational values for every \( x \in \mathbb{R} \). This means that for any real number input, the output is a rational number.

**Hint**: Recall that a continuous function must not have any jumps or breaks in its graph.

2. **Given Condition**:
We are given that \( f(1) + f(2) = 6 \). Let's denote \( f(1) = a \) and \( f(2) = b \). Therefore, we have:
\[ a + b = 6 \]

**Hint**: This equation will help us find the relationship between \( f(1) \) and \( f(2) \).

3. **Continuity and Rational Values**:
Since \( f(x) \) is continuous and takes only rational values, we can infer that \( f(x) \) must be a constant function. This is because if it were not constant, there would be irrational values between any two rational values due to the density of irrational numbers in the real number line.

**Hint**: Think about the implications of continuity on the values that a function can take.

4. **Assume Constant Value**:
Let's assume \( f(x) = c \) for some constant rational number \( c \). Then, we can write:
\[ f(1) = c \quad \text{and} \quad f(2) = c \]
Substituting into the equation \( a + b = 6 \):
\[ c + c = 6 \implies 2c = 6 \implies c = 3 \]

**Hint**: Use the properties of equations to solve for the constant value.

5. **Find \( f(3) \)**:
Since \( f(x) \) is constant and we have found \( c = 3 \), it follows that:
\[ f(3) = c = 3 \]

**Hint**: The value of \( f(x) \) remains the same for all \( x \).

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