To find the domain of the rational function [tex]\( R(x) = \frac{6x}{x-5} \)[/tex], we need to determine the values of [tex]\( x \)[/tex] for which the function is defined. A rational function is defined for all real numbers except where the denominator is zero, as division by zero is undefined.
### Step-by-Step Solution
1. Identify the denominator of the function:
The denominator of [tex]\( R(x) = \frac{6x}{x-5} \)[/tex] is [tex]\( x - 5 \)[/tex].
2. Set the denominator equal to zero:
To find the values of [tex]\( x \)[/tex] that make the denominator zero, solve the equation:
[tex]\[
x - 5 = 0
\][/tex]
3. Solve for [tex]\( x \)[/tex]:
[tex]\[
x - 5 = 0 \implies x = 5
\][/tex]
4. Determine the domain:
The value [tex]\( x = 5 \)[/tex] makes the denominator zero, and thus, [tex]\( x = 5 \)[/tex] is not included in the domain of the function.
Therefore, the domain of [tex]\( R(x) \)[/tex] is all real numbers except [tex]\( x = 5 \)[/tex].
### Conclusion
The correct choice is:
A. The domain of [tex]\( R(x) \)[/tex] is [tex]\(\{x \mid x \neq 5\}\)[/tex].
