find the domain and use set builder notation, y=$$ \sqrt{x^2-25} $$/3x h

To find the domain of the function $ y = \frac{\sqrt{x^2 - 25}}{3x} $, we need to analyze both the square root and the denominator.

### Step 1: Analyzing the square root
The expression inside the square root $ x^2 - 25 $ must be non-negative for the function to be defined (since the square root of a negative number is not real). Therefore, we solve the inequality:
\[
x^2 - 25 \geq 0
\]
\[
x^2 \geq 25
\]
\[
x \leq -5 \quad \text{or} \quad x \geq 5
\]
Thus, $ x \in (-\infty, -5] \cup [5, \infty) $ for the square root to be defined.

### Step 2: Analyzing the denominator
The denominator is $ 3x $, which means $ x \neq 0 $ to avoid division by zero.

### Step 3: Combining the conditions
The domain of the function is the intersection of the two conditions:
- $ x \in (-\infty, -5] \cup [5, \infty) $ from the square root.
- $ x \neq 0 $ from the denominator.

Thus, the domain is:
\[
x \in (-\infty, -5] \cup [5, \infty)
\]

### Set-builder notation:
\[
\{x \,|\, x \leq -5 \text{ or } x \geq 5\}
\]

Would you like more details or have any questions?

---

Here are 5 related questions for practice:
1. What is the domain of $ y = \frac{\sqrt{x^2 - 36}}{4x} $?
2. How do you solve inequalities involving square roots and quadratic expressions?
3. What is the domain of $ y = \frac{1}{\sqrt{x^2 - 25}} $?
4. How would the domain change if $ y = \frac{\sqrt{25 - x^2}}{x} $?
5. What is the range of the function $ y = \frac{\sqrt{x^2 - 25}}{3x} $?

**Tip**: Always check for both square roots and denominators to find the domain of a function.

find the domain and use set builder notation, y=$ \frac{\sqrt{x^2 - 25}}{3x} $

Learn how to find the domain of the function y=$ \frac{\sqrt{x^2 - 25}}{3x} $ using set builder notation. This detailed solution covers key algebraic concepts such as analyzing square roots and denominators, making it suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation