https://file.aiquickdraw.com/mathbot/file/1725927497825ca9lv3bz.jpg

Let's analyze each function shown in the image to determine if it is **even**, **odd**, or **neither**:

1. **Function \( r \) (Top-left graph)**:
   - The function is symmetric with respect to the \( y \)-axis. This means that for any \( x \), the function has \( f(x) = f(-x) \).
   - This is the defining property of an **even** function.
   - **Answer**: Even

2. **Function \( s \) (Top-right graph)**:
   - The function is neither symmetric with respect to the \( y \)-axis nor the origin. It doesn't satisfy either \( f(x) = f(-x) \) (even) or \( f(x) = -f(-x) \) (odd).
   - **Answer**: Neither

3. **Function \( g(x) = -6x^4 + 5x^2 \) (Bottom-left)**:
   - Both terms in the expression have even powers of \( x \) (4 and 2), which means for any \( x \), \( g(-x) = g(x) \).
   - This satisfies the condition for an **even** function.
   - **Answer**: Even

4. **Function \( h(x) = 4x^5 \) (Bottom-right)**:
   - The term has an odd power of \( x \) (5), and \( h(-x) = -h(x) \), which is the defining property of an **odd** function.
   - **Answer**: Odd

### Summary:
- Function \( r \): **Even**
- Function \( s \): **Neither**
- Function \( g(x) = -6x^4 + 5x^2 \): **Even**
- Function \( h(x) = 4x^5 \): **Odd**

Let me know if you want more details or have any questions!

Here are five follow-up questions you might find interesting:
1. What is the general rule for identifying odd and even functions from their algebraic expressions?
2. Can a function be both even and odd? If so, what kind of function would that be?
3. How can the symmetry of a graph help in determining if a function is even or odd?
4. What would the graph of a function that is both even and odd look like?
5. Why do even functions have symmetry around the \( y \)-axis, and odd functions have rotational symmetry around the origin?

**Tip**: Always test a function by plugging in \( f(-x) \) and comparing it to \( f(x) \) to determine if it’s even, odd, or neither!

Determine if the four given functions are even, odd, or neither based on their definitions or graphs.

Explore how to determine whether functions are even, odd, or neither using both graphical symmetry and algebraic expressions. This guide provides a thorough analysis with examples, covering key concepts and formulas related to function symmetry.

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