Question

find the domain.
$f(x)=\frac{7x^2 - 38}{3x + 19}$
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\\}\\)\\(\text{(simplify your answer.)}\\)
\\(\bigcirc\\) c. \\(\\{x|x\text{ is a real number and }x\
eq\square\text{ and }x\
eq\square\\}\\)\\(\text{(simplify your answer. use ascending order.)}\\)
the domain of \\(f\\) in interval notation is \\(\square\\).\\(\text{(use integers or fractions for any numbers in the expression.)}\\)

Explanation:

Step1: Identify the restriction

For a rational function \( f(x)=\frac{7x^2 - 38}{3x + 19} \), the denominator cannot be zero. So we set the denominator equal to zero and solve for \( x \): \( 3x+19 = 0 \).

Step2: Solve for \( x \)

Subtract 19 from both sides: \( 3x=-19 \). Then divide by 3: \( x = -\frac{19}{3} \). So \( x \) cannot be \( -\frac{19}{3} \).

Step3: Determine the domain

The domain is all real numbers except \( x = -\frac{19}{3} \). In set - builder notation, this is \( \{x|x\text{ is a real number and }x
eq-\frac{19}{3}\} \). For the interval notation, we consider all real numbers except \( x = -\frac{19}{3} \), so it is \( (-\infty,-\frac{19}{3})\cup(-\frac{19}{3},\infty) \).

Answer:

For set - builder notation: Option B with \( x
eq-\frac{19}{3} \)
For interval notation: \( (-\infty,-\frac{19}{3})\cup(-\frac{19}{3},\infty) \)