Find the domain of the following functions:
`(x^(2)+5x+30)/(x^(2)-8x+7)`

Options

A. R - {1,7}
B. R - [1,7]
C. R
D. R - {1}

Answer

To find the domain of the function \( f(x) = \frac{x^2 + 5x + 30}{x^2 - 8x + 7} \), 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-by-step solution:

1. **Identify the denominator**:
The denominator of the function is \( q(x) = x^2 - 8x + 7 \).

2. **Set the denominator equal to zero**:
To find the values of \( x \) that make the function undefined, we solve the equation:
\[ x^2 - 8x + 7 = 0 \]

3. **Factor the quadratic equation**:
We can factor the quadratic expression:
\[ x^2 - 8x + 7 = (x - 7)(x - 1) = 0 \]

4. **Find the roots**:
Setting each factor equal to zero gives us the roots:
\[ x - 7 = 0 \quad \Rightarrow \quad x = 7 \]
\[ x - 1 = 0 \quad \Rightarrow \quad x = 1 \]

5. **Determine the domain**:
The function is undefined at \( x = 1 \) and \( x = 7 \). Therefore, the domain of the function is all real numbers except these two points:
\[ \text{Domain of } f(x) = \mathbb{R} - \{1, 7\} \]

### Final answer:
The domain of the function \( f(x) = \frac{x^2 + 5x + 30}{x^2 - 8x + 7} \) is \( \mathbb{R} - \{1, 7\} \).