Question

find the domain of the function.\\( f(x) = \frac{x^3 - x^2 + x + 3}{x^2 + 10x + 21} \\)\
write the domain in set - builder notation. select the correct choice below and, if necessary, fill in any answer box(es) to complete your choice.\
\\( \bigcirc \\) a. \\( \\{x|x \text{ is a real number}\\}\
\\( \bigcirc \\) b. \\( \\{x|x \text{ is a real number and } x \
eq \square\\}\
\quad\quad\quad\text{(simplify your answer.)}\
\\( \bigcirc \\) c. \\( \\{x|x \text{ is a real number and } x \
eq \square \text{ and } x \
eq \square\\}\
\quad\quad\quad\text{(simplify your answer. use ascending order.)}\
the domain in interval notation is \\( \square \\).\
\text{(type your answer in interval notation.)}

Explanation:

Step1: Identify the denominator

The function is a rational function, so we need to find where the denominator is zero because division by zero is undefined. The denominator is \(x^2 + 10x + 21\).

Step2: Factor the denominator

Factor the quadratic expression: \(x^2 + 10x + 21=(x + 3)(x + 7)\).

Step3: Solve for zero

Set each factor equal to zero: \(x + 3 = 0\) gives \(x=-3\), and \(x + 7 = 0\) gives \(x=-7\).

Step4: Determine the domain

The domain of a rational function is all real numbers except the values that make the denominator zero. So we exclude \(x=-7\) and \(x=-3\).

Answer:

C. \(\{x|x\text{ is a real number and }x
eq -7\text{ and }x
eq -3\}\)
Interval notation: \((-\infty, -7)\cup(-7, -3)\cup(-3, \infty)\)