https://file.aiquickdraw.com/mathbot/file/1724002117321il144236.jpg

Let's go through the problems one by one to determine if each represents a function. A relation is a function if every input \( x \) has exactly one output \( y \).

1. **Problem 3**: \( f(x) = x^4 \)
   - This is a polynomial function, specifically a quartic function. For every input \( x \), there is exactly one output \( y \). 
   - **Conclusion**: It **is** a function.

2. **Problem 4**: \( f(x) = |2x + 9| \)
   - This is an absolute value function. Again, for every input \( x \), there is a unique output \( y \).
   - **Conclusion**: It **is** a function.

3. **Problem 5**: \( x = y^2 \)
   - This is a relation where \( y \) is squared to produce \( x \). However, if \( y = \sqrt{x} \), \( y \) could be either positive or negative. For example, \( x = 4 \) could correspond to \( y = 2 \) or \( y = -2 \). 
   - **Conclusion**: It is **not** a function.

4. **Problem 6**: \( y = \sqrt[3]{x} \)
   - This is a cube root function. For every input \( x \), there is one output \( y \). 
   - **Conclusion**: It **is** a function.

5. **Problem 7**: \( y = \frac{6}{x + 3} \)
   - This is a rational function. For every input \( x \) (except \( x = -3 \), which would make the denominator zero and is undefined), there is a unique output \( y \).
   - **Conclusion**: It **is** a function.

6. **Problem 8**: \( f(x) = -x \)
   - This is a linear function. For every input \( x \), there is exactly one output \( y \).
   - **Conclusion**: It **is** a function.

### Summary:
- **Functions**: 3, 4, 6, 7, 8
- **Not a Function**: 5

Would you like a detailed explanation of any specific problem or further help with anything else?

### Follow-up Questions:
1. Why is the relation \( x = y^2 \) not considered a function?
2. What makes a cube root function always a function?
3. How does the absolute value function behave with different inputs?
4. Can a polynomial function ever not be a function?
5. What happens to the output of \( f(x) = -x \) as \( x \) increases?

### Tip:
Remember, a vertical line test can be used graphically to determine if a relation is a function: if a vertical line crosses the graph more than once, it is not a function.

Explore detailed solutions to determine if given equations represent functions. Includes analysis of polynomial functions, absolute value functions, rational functions, and linear functions with clear conclusions based on mathematical criteria.

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