Question

Find the domain and range of the function. $$ g(x)=\frac{2}{x-1} $$

Short answer

The domain of the function \( g(x) = \frac{2}{x - 1} \) is \( (-\infty, 1) \cup (1, \infty) \) and the range is \( (-\infty, \infty) \).

Step-by-step solution

Identifying the Domain

Identify any values of x that would make the denominator zero. In this case, \( x - 1 = 0 \) would make the denominator zero. Solving for x, we get \( x = 1 \). Since x cannot be 1, the domain of the function is all real numbers except 1, which can be written as: \( (-\infty, 1) \cup (1, \infty) \).

Identify the Range

The range of a rational function \( y = \frac{a}{x - h} + k \) is all real numbers except for k. In this case, the function is in the form \( y = \frac{a}{x - 1} \). Therefore, the range of the function is all real numbers. Hence, the range is \( (-\infty, \infty) \).