is y= square root of x^2-4 a function or not

The given equation is:

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

To determine if it represents a function, we need to verify if each value of \( x \) produces a unique corresponding value of \( y \). A function assigns exactly **one output** for every valid input.

### Analysis:

1. **Domain**: 
   The expression inside the square root, \( x^2 - 4 \), must be non-negative because the square root of a negative number is undefined in the set of real numbers. Therefore, we require:
   \[
   x^2 - 4 \geq 0
   \]
   Solving this:
   \[
   x^2 \geq 4
   \]
   So, \( x \geq 2 \) or \( x \leq -2 \). This means the domain of the function is \( x \in (-\infty, -2] \cup [2, \infty) \).

2. **Range**: 
   The square root function always produces non-negative values. Therefore, \( y \geq 0 \).

3. **Uniqueness of Output**: 
   Since we are using the principal (positive) square root, for every valid \( x \) in the domain, there is **only one value** of \( y \). Hence, this equation does assign a unique value of \( y \) for each \( x \).

### Conclusion:
Yes, \( y = \sqrt{x^2 - 4} \) **is a function** because for every valid \( x \), there is only one corresponding \( y \) value.

Would you like more details on the domain, range, or how this function behaves graphically?

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Here are 5 related questions to explore further:
1. What is the range of the function \( y = \sqrt{x^2 - 4} \)?
2. How do you determine the critical points for this function?
3. Can this function be rewritten in a different form?
4. What would happen if the square root included a negative sign, as in \( y = -\sqrt{x^2 - 4} \)?
5. How does this function behave near \( x = 2 \) and \( x = -2 \)?

**Tip**: Always check the domain of a function when dealing with square roots or other operations that impose restrictions!

Is y = square root of x^2 - 4 a function or not?

Explore whether y = √(x^2 - 4) is a function. Learn about the domain, range, and unique output analysis using the definition of a function. This step-by-step solution covers square roots, the domain of the function, and more. Suitable for students in Grades 9-12.

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