Question

find the domain of the function $f(x) = \frac{5}{x - 2}$.\
\
$\circ \\{x \mid x < 2\\}$\
$\circ \\{x \mid x \
eq 2\\}$\
$\circ \\{x \mid x > 2\\}$\
\
express the solution interval notation \
\
question help: video written example

Explanation:

Step1: Identify the restriction

For a rational function \( f(x)=\frac{5}{x - 2} \), the denominator cannot be zero. So we set the denominator equal to zero and solve for \( x \):
\( x - 2=0 \)

Step2: Solve for \( x \)

Adding 2 to both sides of the equation \( x - 2 = 0 \), we get \( x=2 \). This means \( x = 2 \) makes the denominator zero, so it is not in the domain.

Step3: Determine the domain

The domain of the function is all real numbers except \( x = 2 \), which can be written as \( \{x|x
eq2\} \).

Step4: Write in interval notation

To express the domain in interval notation, we consider two intervals: \( (-\infty, 2) \) and \( (2, \infty) \). Combining these, we get \( (-\infty, 2)\cup(2, \infty) \).

Answer:

The correct option for the domain is \( \{x|x
eq2\} \). The interval notation for the domain is \( (-\infty, 2)\cup(2, \infty) \).