Find the domain of the function
`f(x)=(x^(2)+1)//(x^(2)-3x+3)`.

Options

A. `R-{1,2}`
B. `R-{1,4}`
C. R
D. `R-{1}`

Explanation

To find the domain of the function \( f(x) = \frac{x^2 + 1}{x^2 - 3x + 3} \), we need to determine the values of \( x \) for which the function is defined. The function is defined as long as the denominator is not equal to zero.

### Step 1: Identify the denominator
The denominator of the function is \( x^2 - 3x + 3 \).

### Step 2: Set the denominator equal to zero
To find the values of \( x \) that make the function undefined, we set the denominator equal to zero:
\[ x^2 - 3x + 3 = 0 \]

### Step 3: Calculate the discriminant
We will use the discriminant to determine if there are any real roots. The discriminant \( D \) is given by:
\[ D = b^2 - 4ac \]
where \( a = 1 \), \( b = -3 \), and \( c = 3 \). Thus,
\[ D = (-3)^2 - 4 \cdot 1 \cdot 3 = 9 - 12 = -3 \]

### Step 4: Analyze the discriminant
Since the discriminant \( D \) is negative, this means that the quadratic equation \( x^2 - 3x + 3 = 0 \) has no real roots. Therefore, the denominator is never zero for any real value of \( x \).

### Step 5: Conclusion about the domain
Since the denominator is never zero, the function is defined for all real numbers. Thus, the domain of the function \( f(x) \) is:
\[ \text{Domain of } f(x) = \mathbb{R} \]

### Final Answer:
The domain of the function \( f(x) = \frac{x^2 + 1}{x^2 - 3x + 3} \) is \( \mathbb{R} \).

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