To determine whether the function $f(x) = 5x - 4x^2$ is even, odd, or neither, we follow these steps:

Step 1: Check if the function is even

A function $f(x)$ is even if: $f(-x) = f(x)$ Substitute $-x$ into the function: $f(-x) = 5(-x) - 4(-x)^2$ Simplify the expression: $f(-x) = -5x - 4x^2$ Now, compare this result with $f(x) = 5x - 4x^2$: $f(-x) = -5x - 4x^2 \quad \text{and} \quad f(x) = 5x - 4x^2$ Since $f(-x) \neq f(x)$, the function is not even.

Step 2: Check if the function is odd

A function $f(x)$ is odd if: $f(-x) = -f(x)$ Substitute $f(x) = 5x - 4x^2$ into $-f(x)$: $-f(x) = -(5x - 4x^2) = -5x + 4x^2$ Now compare $f(-x)$ and $-f(x)$: $f(-x) = -5x - 4x^2 \quad \text{and} \quad -f(x) = -5x + 4x^2$ Since $f(-x) \neq -f(x)$, the function is not odd.

Conclusion:

The function $f(x) = 5x - 4x^2$ is neither even nor odd because it satisfies neither of the conditions for being even or odd.

Would you like more details on this process or have any other questions?

Here are 5 related questions:

1. How do you determine if a polynomial function is even or odd?
2. What is the difference between even and odd functions geometrically?
3. Can a function be both even and odd?
4. What are some common examples of even and odd functions?
5. How does symmetry in functions relate to evenness and oddness?

Tip: Even functions are symmetric about the y-axis, while odd functions are symmetric about the origin.