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Identify if the given function is even, odd, neither, or both.



To determine if the given function is even, odd, neither, or both, we can analyze the symmetry of the graph.

### **Even Function:**
A function is even if its graph is symmetric with respect to the y-axis. This means that for every point \((x, y)\) on the graph, there is a corresponding point \((-x, y)\) on the graph as well. Mathematically, \( f(x) = f(-x) \).

### **Odd Function:**
A function is odd if its graph is symmetric with respect to the origin. This means that for every point \((x, y)\) on the graph, there is a corresponding point \((-x, -y)\) on the graph as well. Mathematically, \( f(x) = -f(-x) \).

### **Analysis of the Graph:**
- **Symmetry about the y-axis:** The graph does **not** exhibit symmetry about the y-axis.
- **Symmetry about the origin:** The graph does **not** exhibit symmetry about the origin.

Since the graph does not show symmetry about the y-axis (even function) or the origin (odd function), the function is **neither** even nor odd.

Would you like more details or have any further questions?

### Related Questions:
1. How do you determine the evenness or oddness of a function algebraically?
2. What are some examples of even and odd functions?
3. Can a function be both even and odd? If so, under what conditions?
4. How does the degree of a polynomial affect its symmetry properties?
5. What are the implications of a function being neither even nor odd?
6. How would the graph change if the function were odd?
7. How can transformations (like translations) affect the symmetry of a graph?
8. What does the end behavior of this graph suggest about the function's degree?

### Tip:
When analyzing the symmetry of a graph, always check the symmetry with respect to the y-axis for even functions and with respect to the origin for odd functions.

Understand how to determine if a function is even, odd, neither, or both through graphical analysis. Explore the concepts of function symmetry, even functions, and odd functions with detailed examples. Suitable for high school students and anyone interested in mathematical function properties.

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