For each equation below, determine if the function is Odd, Even, or Neither

f(x)=2x-x3



Odd

Even

Neither


g(x)=x



Odd

Even

Neither


h(x)=2x4



Odd

Even

Neither

To determine if a function is **odd**, **even**, or **neither**, we follow these rules:

- **Even function**: \( f(-x) = f(x) \)
- **Odd function**: \( f(-x) = -f(x) \)
- **Neither**: If neither of the above conditions hold.

### Let's analyze each function:

1. **For** \( f(x) = 2x - x^3 \):
   - Compute \( f(-x) \):
     \[
     f(-x) = 2(-x) - (-x)^3 = -2x + x^3
     \]
   - Compare this with \( -f(x) \):
     \[
     -f(x) = -(2x - x^3) = -2x + x^3
     \]
   - Since \( f(-x) = -f(x) \), the function is **odd**.

2. **For** \( g(x) = x \):
   - Compute \( g(-x) \):
     \[
     g(-x) = -x
     \]
   - Compare this with \( -g(x) \):
     \[
     -g(x) = -x
     \]
   - Since \( g(-x) = -g(x) \), the function is **odd**.

3. **For** \( h(x) = 2x^4 \):
   - Compute \( h(-x) \):
     \[
     h(-x) = 2(-x)^4 = 2x^4
     \]
   - Since \( h(-x) = h(x) \), the function is **even**.

### Final answers:
- \( f(x) = 2x - x^3 \) is **odd**.
- \( g(x) = x \) is **odd**.
- \( h(x) = 2x^4 \) is **even**.

Would you like more details on any of these calculations?

### 5 Related Questions:
1. What is the geometric interpretation of odd and even functions?
2. How can you determine if a polynomial function is odd, even, or neither by looking at its degree?
3. How does symmetry about the origin relate to odd functions?
4. Can a function be both odd and even? Why or why not?
5. How does a function’s graph change based on whether it is odd, even, or neither?

**Tip:** If all powers in a polynomial are odd, the function is likely odd, and if all powers are even, it's likely even.

For each equation below, determine if the function is Odd, Even, or Neither:

f(x) = 2x - x^3
Odd
Even
Neither

g(x) = x
Odd
Even
Neither

h(x) = 2x^4
Odd
Even
Neither

Learn how to determine if functions are odd, even, or neither using symmetry properties. This solution covers detailed steps for analyzing the functions f(x) = 2x - x^3, g(x) = x, and h(x) = 2x^4, with explanations on odd and even function properties, suitable for students in Grades 10-12.

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