https://file.aiquickdraw.com/mathbot/file/1734444668996pks7rgz2.jpg

To determine whether each function is *even*, *odd*, or *neither*, let's analyze them one by one based on the provided graphs and equations.

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### 1. **Function \( r \) (Top Left Graph)**  
A function is **even** if it satisfies \( f(-x) = f(x) \) and **odd** if \( f(-x) = -f(x) \).  
- Observing the graph, the function \( r \) is symmetric about the **y-axis**.  
- This symmetry indicates that \( f(-x) = f(x) \), which is the definition of an **even function**.

**Answer:** Even  

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### 2. **Function \( s \) (Top Right Graph)**  
- To determine symmetry, we check if the function has rotational symmetry about the origin (odd) or y-axis symmetry (even).  
- Observing the graph, the function \( s \) has **no symmetry** about the y-axis or the origin.  
- Therefore, it is **neither even nor odd**.

**Answer:** Neither  

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### 3. **Function \( g(x) = -6x^4 + 7x^2 \)**  
- For an algebraic function:  
  - A function is **even** if all the exponents of \( x \) are even.  
  - A function is **odd** if all the exponents of \( x \) are odd.  
- In \( g(x) = -6x^4 + 7x^2 \), all powers of \( x \) (4 and 2) are even.  
- This confirms that \( g(x) \) is **even**.

**Answer:** Even  

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### 4. **Function \( h(x) = -4x^5 + 7x^2 \)**  
- In \( h(x) = -4x^5 + 7x^2 \), we see:  
  - The term \( -4x^5 \) has an odd power (5).  
  - The term \( 7x^2 \) has an even power (2).  
- Since \( h(x) \) contains a mix of odd and even powers, it cannot satisfy the conditions for being purely even or odd.  
- Therefore, \( h(x) \) is **neither even nor odd**.

**Answer:** Neither  

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### Final Answers:
1. Function \( r \): **Even**  
2. Function \( s \): **Neither**  
3. Function \( g(x) \): **Even**  
4. Function \( h(x) \): **Neither**  

Let me know if you'd like further clarification on any of these!

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### Related Questions:  
1. What is the algebraic test to check if a function is odd or even?  
2. How does symmetry about the origin relate to odd functions?  
3. Can a function be both even and odd? Why or why not?  
4. How do polynomial exponents determine whether a function is even or odd?  
5. How can transformations (e.g., shifts or reflections) affect function symmetry?  

### Tip:  
**Even functions** have symmetry about the y-axis, while **odd functions** have rotational symmetry about the origin. Graphing can provide a quick visual confirmation!

Determine whether the given functions r, s, g(x), and h(x) are even, odd, or neither.

Learn how to determine whether a function is even, odd, or neither using symmetry and algebraic analysis. This guide covers graph-based symmetry checks for functions r and s, as well as polynomial functions g(x) = -6x^4 + 7x^2 and h(x) = -4x^5 + 7x^2. Suitable for high school students studying algebra and function properties.

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