In the context of the equation \( y = 2^x \):

1. **Definition of Exponential Functions:**
   - Exponential functions with a positive base greater than 1, such as \( 2^x \), always produce positive results for all real values of \( x \).
   - The function \( 2^x \) represents exponential growth, and its output (range) is always positive.

2. **Mathematical Justification:**
   - For any real number \( x \), \( 2^x \) is defined to be greater than 0. Specifically:
     \[
     2^x > 0 \text{ for all } x \in \mathbb{R}
     \]

Therefore, when solving the quadratic equation \( y^2 - 2y - 8 = 0 \) and finding the roots \( y = 4 \) and \( y = -2 \), we discard \( y = -2 \) because:

- \( y = 2^x \) implies \( 2^x = -2 \).
- Since \( 2^x \) can never be negative, \( y = -2 \) is not a valid solution.

This is why \( y \) cannot be negative in this context.

Would you like more details or have any questions about this explanation?

### Related Questions:
1. Why is the range of exponential functions always positive?
2. How does the base of the exponent affect the range of the function?
3. What are the key properties of exponential functions?
4. How do you solve equations involving exponential functions with different bases?
5. What is the relationship between exponential and logarithmic functions?
6. Can exponential functions ever have zero as a value?
7. How do you graph an exponential function like \( y = 2^x \)?
8. What are the applications of exponential functions in real-world problems?

### Tip:
When working with exponential equations, always remember that the range of an exponential function with a positive base is always positive. This can help you quickly identify and discard invalid solutions.

Understand why the variable y cannot be negative in the context of exponential functions like y = 2^x. Explore the mathematical reasoning behind the positivity of exponential results and why y = -2 is not a valid solution in equations involving exponential growth.

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