Option D:
[tex]\{x: x \neq 1\}[/tex] is the domain of the function.
Solution:
Given function is
[tex]$r(x)=\frac{2 x}{x-1}[/tex]
To find the domain of the function:
Option A: [tex]\{x: x \neq 0\}[/tex]
Substitute x = 0 in r(x).
[tex]$r(0)=\frac{2 \times 0}{0-1}=0[/tex]
If x = 0, then r(0) = 0
So that x ≠ 0 is false.
So, [tex]\{x: x \neq 0\}[/tex] is not the domain of the function.
Option B: [tex]\{x: x \neq \pm 1\}[/tex]
Substitute x = –1 in r(x).
[tex]$r(-1)=\frac{2 \times (-1)}{-1-1}=1[/tex]
If x = –1, then r(–1) = 1
So that x = ± 1 is false.
So, [tex]\{x: x \neq \pm 1\}[/tex] is not the domain of the function.
Option C: [tex]$\{x: \text { all real numbers }\}$[/tex]
Substitute x = 1 in r(x).
[tex]$r(1)=\frac{2 \times 1}{1-1}=\frac{2}{0}[/tex]
It is indeterminate.
So, all real numbers are not the domain of the function.
Option D: [tex]\{x: x \neq 1\}[/tex]
Substitute x = 1 in r(x).
[tex]$r(1)=\frac{2 \times 1}{1-1}=\frac{2}{0}[/tex]
It is indeterminate.
So, [tex]\{x: x \neq 1\}[/tex] is the domain of the function.
