https://file.aiquickdraw.com/mathbot/file/1726872325611drnwr2ps.jpg

To determine whether the function \( m(x) = -5x^5 + 4x^3 + x \) is even, odd, or neither, let's apply the following steps:

1. **For an even function**, the rule is:
   \[
   m(-x) = m(x)
   \]
2. **For an odd function**, the rule is:
   \[
   m(-x) = -m(x)
   \]
3. If neither of these conditions hold, the function is **neither even nor odd**.

### Step-by-step analysis:

1. **Calculate \( m(-x) \):**
   \[
   m(-x) = -5(-x)^5 + 4(-x)^3 + (-x)
   \]
   Simplifying:
   \[
   m(-x) = -5(-x^5) + 4(-x^3) + (-x) = 5x^5 - 4x^3 - x
   \]

2. **Compare \( m(-x) \) with \( m(x) \):**
   - \( m(x) = -5x^5 + 4x^3 + x \)
   - \( m(-x) = 5x^5 - 4x^3 - x \)

   Notice that \( m(-x) \) is the **negative** of \( m(x) \). This satisfies the condition for an odd function.

### Conclusion:
The function \( m(x) = -5x^5 + 4x^3 + x \) is an **odd function**.

Would you like further clarification or details on this?

### Related questions:
1. What are the key characteristics of even functions?
2. How do you determine if a function is neither even nor odd?
3. Can a function be both even and odd?
4. How do polynomial degrees influence whether a function is even or odd?
5. Are there any graphical interpretations of even and odd functions?

### Tip:
Remember that the powers of the terms in a polynomial help determine the function's symmetry—odd powers suggest odd symmetry, while even powers suggest even symmetry.

Determine whether the function m(x) = -5x^5 + 4x^3 + x is even, odd, or neither.

Learn how to determine if the function m(x) = -5x^5 + 4x^3 + x is even, odd, or neither. This problem covers polynomial symmetry and function classification. Suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation