What is the domain of the function 
�
=
2
�
5
�
2
−
4
y= 
5x 
2
 −4
​
 
2x
​
 ?

The function in question is:

\[
y = \frac{2x}{\sqrt{5x^2 - 4}}
\]

To find the domain, we need to consider the following conditions:

### 1. **Square Root Condition:**
   The expression inside the square root must be non-negative, because the square root of a negative number is not a real number. Therefore, we need:
   \[
   5x^2 - 4 \geq 0
   \]

### 2. **Solve the Inequality:**
   \[
   5x^2 \geq 4
   \]
   Divide both sides by 5:
   \[
   x^2 \geq \frac{4}{5}
   \]
   Taking the square root on both sides:
   \[
   |x| \geq \sqrt{\frac{4}{5}} = \frac{2}{\sqrt{5}} = \frac{2\sqrt{5}}{5}
   \]
   This implies:
   \[
   x \leq -\frac{2\sqrt{5}}{5} \quad \text{or} \quad x \geq \frac{2\sqrt{5}}{5}
   \]

### 3. **Denominator Condition:**
   The denominator cannot be zero because division by zero is undefined. Therefore, we must have:
   \[
   5x^2 - 4 \neq 0
   \]
   Which simplifies to:
   \[
   x^2 \neq \frac{4}{5}
   \]
   This does not change our interval but confirms the boundary values are excluded.

### Conclusion:
The domain of the function \( y = \frac{2x}{\sqrt{5x^2 - 4}} \) is:

\[
(-\infty, -\frac{2\sqrt{5}}{5}) \cup (\frac{2\sqrt{5}}{5}, \infty)
\]

This excludes the interval \(-\frac{2\sqrt{5}}{5} \leq x \leq \frac{2\sqrt{5}}{5}\), where the expression inside the square root would be non-positive.

Would you like more details, or do you have any other questions?

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Here are 5 related questions:

1. What is the domain of the function \( y = \frac{1}{\sqrt{9x^2 - 16}} \)?
2. How does the domain change if the function is \( y = \frac{3x + 1}{\sqrt{7 - x^2}} \)?
3. What happens to the domain if the square root is in the numerator instead?
4. How would you determine the domain if the expression inside the square root is a cubic polynomial?
5. Can you determine the domain of a function with a square root in both the numerator and denominator?

**Tip:** When solving inequalities involving square roots, it's important to remember that the expression under the square root must be greater than zero when in the denominator.

Explore how to determine the domain of a function containing square roots with a detailed example. Learn about the conditions for the domain involving square roots and how to apply them in different scenarios. Suitable for advanced high school students and college-level learners.

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