how to determine if a function is even or odd

To determine whether a function is even, odd, or neither, you can follow these steps:

### 1. **Even Function Test:**
   A function \( f(x) \) is **even** if it satisfies the condition:
   \[
   f(-x) = f(x)
   \]
   This means that the function's graph is symmetric about the y-axis.

### 2. **Odd Function Test:**
   A function \( f(x) \) is **odd** if it satisfies the condition:
   \[
   f(-x) = -f(x)
   \]
   This means that the function's graph is symmetric about the origin.

### Steps to check:
- **Step 1**: Substitute \( -x \) into the function, i.e., find \( f(-x) \).
- **Step 2**: Compare \( f(-x) \) with \( f(x) \) and \( -f(x) \).
   - If \( f(-x) = f(x) \), the function is even.
   - If \( f(-x) = -f(x) \), the function is odd.
   - If neither condition holds, the function is neither even nor odd.

### Example 1: \( f(x) = x^2 \)
- \( f(-x) = (-x)^2 = x^2 \)
- Since \( f(-x) = f(x) \), the function is even.

### Example 2: \( f(x) = x^3 \)
- \( f(-x) = (-x)^3 = -x^3 \)
- Since \( f(-x) = -f(x) \), the function is odd.

### Example 3: \( f(x) = x^2 + x \)
- \( f(-x) = (-x)^2 + (-x) = x^2 - x \)
- Since \( f(-x) \neq f(x) \) and \( f(-x) \neq -f(x) \), the function is neither even nor odd.

Would you like more details on this topic?

Here are 5 related questions for deeper understanding:
1. What are examples of neither even nor odd functions?
2. How do you determine if a piecewise function is even or odd?
3. Can a function be both even and odd?
4. How does symmetry in graphs help determine if a function is even or odd?
5. What are the practical applications of even and odd functions in real-world problems?

**Tip**: When checking for symmetry, sketching the graph of the function can help visually confirm whether it is even, odd, or neither.

Understand how to determine whether a function is even, odd, or neither with a clear step-by-step process. Learn about key concepts in algebra such as symmetry in graphs and the properties of even and odd functions. This guide is ideal for students in Grades 9-12.

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