Question

find the domain of the function.\\( f(x) = \frac{x}{x - 5} \\)\
select one:\
a. \\( (-infty, 0) \\)\
b. \\( (-infty, -5) cup (-5, infty) \\)\
c. \\( (0, infty) \\)\
d. \\( (-infty, 5) cup (5, infty) \\)

Explanation:

Step1: Identify the restriction

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

Step2: Solve for \( x \)

Adding 5 to both sides of the equation \( x - 5=0 \), we get \( x = 5 \). This means \( x = 5 \) 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 \( x = 5 \). In interval notation, this is \( (-\infty,5)\cup(5,\infty) \).

Answer:

D. \( (-\infty, 5) \cup (5, \infty) \)