Find the domain of the function. (Enter your answer in interval notation.) 1/$$ ∜x^2-9x $$

We are given the function $ f(x) = \frac{1}{\sqrt[4]{x^2 - 9x}} $, and we need to find its domain.

### Step 1: Analyzing the Denominator
The function has a denominator of $ \sqrt[4]{x^2 - 9x} $. Since this is a root in the denominator, the expression inside the fourth root must be positive, and it cannot be zero because that would make the denominator undefined. Thus, we must ensure that:

\[
x^2 - 9x > 0
\]

### Step 2: Solving the Inequality
We solve the inequality $ x^2 - 9x > 0 $ by factoring:

\[
x(x - 9) > 0
\]

This inequality holds when both $ x $ and $ x - 9 $ have the same sign. The critical points occur at $ x = 0 $ and $ x = 9 $. These divide the real number line into three intervals:

1. $ (-\infty, 0) $
2. $ (0, 9) $
3. $ (9, \infty) $

We will check the sign of $ x(x - 9) $ in each of these intervals.

- In the interval $ (-\infty, 0) $: Choose $ x = -1 $. Then $ x(x - 9) = (-1)(-1 - 9) = (-1)(-10) = 10 $, which is positive.
- In the interval $ (0, 9) $: Choose $ x = 1 $. Then $ x(x - 9) = (1)(1 - 9) = (1)(-8) = -8 $, which is negative.
- In the interval $ (9, \infty) $: Choose $ x = 10 $. Then $ x(x - 9) = (10)(10 - 9) = (10)(1) = 10 $, which is positive.

Thus, the inequality $ x(x - 9) > 0 $ is satisfied in the intervals $ (-\infty, 0) $ and $ (9, \infty) $.

### Step 3: Conclusion
The function is defined where the expression inside the fourth root is positive, which corresponds to the intervals $ (-\infty, 0) $ and $ (9, \infty) $.

Therefore, the domain of the function is:

\[
(-\infty, 0) \cup (9, \infty)
\]

Let me know if you need any details or further clarification!  

### Related Questions:
1. How do you find the domain of a square root function?
2. What happens to the domain when there is an even root in the denominator?
3. How does the behavior of polynomial factors help determine the sign of an inequality?
4. How do we analyze critical points in solving inequalities?
5. What are the implications of having an undefined function at certain points?

**Tip:** When dealing with even roots in the denominator, always ensure the radicand (the expression inside the root) is positive and non-zero.

Learn how to find the domain of the function f(x) = 1/(Fourth Root of x^2 - 9x) with a step-by-step solution. Understand the conditions under which the denominator is defined and explore the intervals where the function is defined. Suitable for advanced high school students and college-level mathematics enthusiasts.

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