Question

find the domain of the function $f(x)=\frac{3x - 8}{x + 8}$. write both set - builder notation and interval notation for the answer

write the domain in set - builder notation. select the correct choice below and, if necessary, fill in the answer box to complete your choice

\\(\bigcirc\\) a. the domain of $f$ is \\(\\{x|x\\) is a real number and $x = \square\\}$.
( type an integer or a fraction. use a comma to separate answers as needed. )

\\(\bigcirc\\) b. the domain of $f$ is \\(\\{x|x\\) is a real number and $x\
eq\square\\}$.
( type an integer or a fraction. use a comma to separate answers as needed. )

\\(\bigcirc\\) c. the domain of $f$ is \\(\\{x|x\\) is a real number\\}.

the domain of $f$ is \\(\square\\).
( type your answer in interval notation. )

Explanation:

Step1: Identify the restriction

For a rational function \( f(x)=\frac{3x - 8}{x + 8} \), the denominator cannot be zero. So we set the denominator equal to zero and solve for \( x \): \( x+8 = 0 \).

Step2: Solve for \( x \)

Subtract 8 from both sides: \( x=-8 \). So \( x
eq - 8 \) because that would make the denominator zero, which is undefined.

Step3: Set - builder notation

The domain in set - builder notation is all real numbers except \( x = - 8 \), so the correct choice is B, and we fill in \( - 8 \) in the box for option B.

Step4: Interval notation

To write the domain in interval notation, we consider all real numbers less than - 8, and all real numbers greater than - 8. So the interval notation is \( (-\infty,-8)\cup(-8,\infty) \).

Answer:

Set - builder notation: B. The domain of \( f \) is \( \{x|x\text{ is a real number and }x
eq - 8\} \)
Interval notation: \( (-\infty,-8)\cup(-8,\infty) \)