To determine the domain of the function $f(x) = \sqrt{\frac{x}{x-1}}$, we need to consider the conditions under which the expression inside the square root is defined and non-negative, because square roots of negative numbers are not real numbers.

Step 1: Identify when the expression inside the square root is defined.

The expression inside the square root is $\frac{x}{x-1}$. For this fraction to be defined, the denominator cannot be zero: $x - 1 \neq 0 \implies x \neq 1$ Thus, $x = 1$ is excluded from the domain.

Step 2: Identify when the expression inside the square root is non-negative.

For the square root to be real, we need: $\frac{x}{x-1} \geq 0$ To solve this inequality, let's analyze the sign of the fraction $\frac{x}{x-1}$.

• The numerator $x$ is positive when $x > 0$ and negative when $x < 0$.
• The denominator $x - 1$ is positive when $x > 1$ and negative when $x < 1$.

Now, let’s find the intervals where $\frac{x}{x-1} \geq 0$. The sign of the fraction depends on the signs of $x$ and $x - 1$. The critical points are $x = 0$ and $x = 1$.

We need to consider the following intervals:

1. $(-\infty, 0)$
2. $(0, 1)$
3. $(1, \infty)$

Interval analysis:

• For $x < 0$: both $x$ and $x - 1$ are negative, so $\frac{x}{x-1} > 0$.
• For $0 < x < 1$: $x$ is positive and $x - 1$ is negative, so $\frac{x}{x-1} < 0$.
• For $x > 1$: both $x$ and $x - 1$ are positive, so $\frac{x}{x-1} > 0$.

Thus, the inequality $\frac{x}{x-1} \geq 0$ holds when: $x \in (-\infty, 0] \cup (1, \infty)$ However, remember that $x = 1$ is excluded from the domain (since the denominator would be zero), so the domain is: $x \in (-\infty, 0] \cup (1, \infty)$

Domain of the function:

The domain of $f(x) = \sqrt{\frac{x}{x-1}}$ is: $(-\infty, 0] \cup (1, \infty)$

Do you want more details or have any questions on this?

Here are some related questions for further exploration:

1. How would the domain change if the expression inside the square root was a different rational function?
2. How can we solve inequalities like $\frac{x}{x-1} \geq 0$ using a sign chart?
3. What happens to the domain if the square root is replaced by a cube root?
4. How can we find the domain of a function involving a logarithm or trigonometric functions?
5. What is the effect of including an absolute value in the function?

Tip: Always check the denominator of rational expressions to avoid division by zero when finding the domain!