Question

Which of the following correctly describes the domain of the function shown

Answer 1

Option D:

$\{x: x \neq 1\}$ is the domain of the function.

Solution:

Given function is

$r(x)=\frac{2 x}{x-1}$

To find the domain of the function:

Option A: $\{x: x \neq 0\}$

Substitute x = 0 in r(x).

$r(0)=\frac{2 \times 0}{0-1}=0$

If x = 0, then r(0) = 0

So that x ≠ 0 is false.

So, $\{x: x \neq 0\}$ is not the domain of the function.

Option B: $\{x: x \neq \pm 1\}$

Substitute x = –1 in r(x).

$r(-1)=\frac{2 \times (-1)}{-1-1}=1$

If x = –1, then r(–1) = 1

So that x = ± 1 is false.

So, $\{x: x \neq \pm 1\}$ is not the domain of the function.

Option C: $\{x: \text { all real numbers }\}$

Substitute x = 1 in r(x).

$r(1)=\frac{2 \times 1}{1-1}=\frac{2}{0}$

It is indeterminate.

So, all real numbers are not the domain of the function.

Option D: $\{x: x \neq 1\}$

Substitute x = 1 in r(x).

$r(1)=\frac{2 \times 1}{1-1}=\frac{2}{0}$

It is indeterminate.

So, $\{x: x \neq 1\}$ is the domain of the function.