Q 46. - Precalculus Enhanced with Graphing Utilities

Question

Find the domain of the function : $h\left(x\right)={\log }_3\left(\frac{x}{x-1}\right)$

Step-by-Step Solution

Answer

The domain of the given function is : $(-\infty ,0)\cup (1,\infty )$

Step 1. Given information

Given expression is $h\left(x\right)={\log }_3\left(\frac{x}{x-1}\right)$

Domain : The range of $x$ is known as domain of definition.

Step 2. Find domain

The domain of definition of the function is the range of the values where $\frac{x}{x-1}>0$.

In this range the function will produce finite values ; $x$ and $(x-1)$ both will be either positive or negative.

Now, $(x-1)$ will be positive in the range $(1,\infty )$ and $x$ will be positive in the range $(0,\infty )$.

Again, $(x-1)$ will be negative in the range $(-\infty ,1)$ and $x$will be negative in the range $(-\infty ,0)$

Thus, the domain of the given function is $(-\infty ,0)\cup (1,\infty )$

Step 3. Check

$\log \left(0\right)$ is undefined.

Hence, the function is discontinued from $0\le x\le 1$

Hence, the domain of the function is $(-\infty ,0)\cup (1,\infty )$