The domain of the function `(x^(2) + 8x + 9)/(x^(2) - 9x + 20)` is

Options

A. R
B. R - {9}
C. R - {20}
D. R - {4, 5}

Answer

To find the domain of the function \( f(x) = \frac{x^2 + 8x + 9}{x^2 - 9x + 20} \), we need to determine the values of \( x \) for which the function is defined. The function will be undefined when the denominator is equal to zero.

### Step-by-Step Solution:

1. **Identify the Denominator**:
The denominator of the function is \( x^2 - 9x + 20 \).

2. **Set the Denominator to Zero**:
To find the values that make the function undefined, we set the denominator equal to zero:
\[ x^2 - 9x + 20 = 0 \]

3. **Factor the Quadratic**:
We will factor the quadratic expression. We need two numbers that multiply to \( 20 \) (the constant term) and add up to \( -9 \) (the coefficient of \( x \)). The numbers \( -4 \) and \( -5 \) satisfy this condition:
\[ x^2 - 9x + 20 = (x - 4)(x - 5) \]

4. **Set Each Factor to Zero**:
Now, we set each factor equal to zero to find the values of \( x \) that make the denominator zero:
\[ x - 4 = 0 \quad \Rightarrow \quad x = 4 \]
\[ x - 5 = 0 \quad \Rightarrow \quad x = 5 \]

5. **Determine the Domain**:
The function is undefined at \( x = 4 \) and \( x = 5 \). Therefore, the domain of the function includes all real numbers except \( 4 \) and \( 5 \):
\[ \text{Domain} = \mathbb{R} \setminus \{4, 5\} \]

### Final Answer:
The domain of the function \( f(x) = \frac{x^2 + 8x + 9}{x^2 - 9x + 20} \) is all real numbers except \( 4 \) and \( 5 \).