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.