Find the domain of the function ,`f(x)=(x^2 +2x +3)/(x^2 -5x +6)`

Answer

To find the domain of the function \( f(x) = \frac{x^2 + 2x + 3}{x^2 - 5x + 6} \), we need to determine the values of \( x \) for which the function is defined. The function will be undefined wherever the denominator is equal to zero.

### Step 1: Set the denominator equal to zero
We start by finding the roots of the denominator:
\[ x^2 - 5x + 6 = 0 \]

### Step 2: Factor the quadratic equation
We can factor the quadratic equation:
\[ x^2 - 5x + 6 = (x - 2)(x - 3) = 0 \]

### Step 3: Solve for \( x \)
Setting each factor equal to zero gives us:
\[ x - 2 = 0 \quad \Rightarrow \quad x = 2 \]
\[ x - 3 = 0 \quad \Rightarrow \quad x = 3 \]

### Step 4: Identify the values to exclude from the domain
The function \( f(x) \) is undefined at \( x = 2 \) and \( x = 3 \). Therefore, these values must be excluded from the domain.

### Step 5: Write the domain in interval notation
The domain of the function is all real numbers except \( 2 \) and \( 3 \). In interval notation, this can be expressed as:
\[ \text{Domain} = (-\infty, 2) \cup (2, 3) \cup (3, \infty) \]

### Final Answer
Thus, the domain of the function \( f(x) \) is:
\[ \text{Domain} = \{ x \in \mathbb{R} \mid x \neq 2, x \neq 3 \} \]

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