If `f : R->Q` (Rational numbers), `g : R ->Q` (Rational numbers) are two continuous functions such that `sqrt(3)f(x)+g(x)=4` then `(1-f(x))^3+(g(x)-3)^3` is equal to (1) `1` (2)`2` (3) `3` (4) `4 `

Options

A. 1
B. 2
C. 3
D. 4

Explanation

To solve the problem, we start with the equation given in the question:

\[ \sqrt{3}f(x) + g(x) = 4 \]

where \( f: \mathbb{R} \to \mathbb{Q} \) and \( g: \mathbb{R} \to \mathbb{Q} \) are continuous functions mapping real numbers to rational numbers.

### Step 1: Analyze the equation
Since \( \sqrt{3} \) is an irrational number and both \( f(x) \) and \( g(x) \) are rational functions, the left-hand side of the equation must also be rational. For the sum \( \sqrt{3}f(x) + g(x) \) to be rational, the term \( \sqrt{3}f(x) \) must be equal to \( 0 \) because the sum of an irrational number and a rational number cannot yield a rational number unless the irrational part is zero.

### Step 2: Set \( f(x) \) to zero
From the analysis above, we can conclude:
\[ \sqrt{3}f(x) = 0 \implies f(x) = 0 \]

### Step 3: Substitute \( f(x) \) into the equation
Now, substituting \( f(x) = 0 \) back into the original equation:
\[ \sqrt{3}(0) + g(x) = 4 \implies g(x) = 4 \]

### Step 4: Calculate the expression
We need to find the value of the expression:
\[ (1 - f(x))^3 + (g(x) - 3)^3 \]
Substituting \( f(x) = 0 \) and \( g(x) = 4 \):
\[ (1 - 0)^3 + (4 - 3)^3 = 1^3 + 1^3 = 1 + 1 = 2 \]

### Final Answer
Thus, the value of \( (1 - f(x))^3 + (g(x) - 3)^3 \) is \( 2 \).

### Conclusion
The correct option is (2) \( 2 \).

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