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

Answer

To find the domain of the function \( f(x) = \frac{x^2 + 5x + 30}{x^2 - 8x + 7} \), we need to identify the values of \( x \) for which the function is defined. Since this is a rational function, it is defined for all real numbers except where the denominator is equal to zero.

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

### 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:
\[ x^2 - 8x + 7 = 0 \]

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

### Step 4: Solve for \( x \)
Setting each factor equal to zero gives us the roots:
\[ x - 1 = 0 \quad \Rightarrow \quad x = 1 \]
\[ x - 7 = 0 \quad \Rightarrow \quad x = 7 \]

### Step 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 values:
\[ \text{Domain} = \mathbb{R} - \{1, 7\} \]

### Final Answer
The domain of the function \( f(x) = \frac{x^2 + 5x + 30}{x^2 - 8x + 7} \) is:
\[ \text{Domain} = (-\infty, 1) \cup (1, 7) \cup (7, \infty) \]

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