Question

find the domain of the function $f(x) = \frac{2x - 5}{x + 6}$. 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 undefined points

For a rational function \( f(x)=\frac{2x - 5}{x + 6} \), the denominator cannot be zero. So, set \( x + 6=0 \).
Solving \( x+6 = 0 \), we get \( x=-6 \).

Step2: Determine set - builder notation

The domain consists of all real numbers except \( x = - 6 \). So, in set - builder notation, the domain is \( \{x|x\text{ is a real number and }x
eq - 6\} \), so we choose option B and fill in \( - 6 \) in the box.

Step3: Determine interval notation

The domain in interval notation is \( (-\infty,-6)\cup(-6,\infty) \), because \( x \) can take any real value less than \( - 6 \), any real value greater than \( - 6 \), but not \( - 6 \) itself.

Answer:

Set - builder notation: The domain of \( f \) is \( \{x|x\text{ is a real number and }x
eq - 6\} \) (corresponding to option B with \( - 6 \) filled).
Interval notation: The domain of \( f \) is \( (-\infty,-6)\cup(-6,\infty) \)