determine algebraically whether f(x)=5x-4x^2 is even, odd or neither

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.

Learn how to algebraically determine whether the function f(x) = 5x - 4x^2 is even, odd, or neither. This solution walks through the key concepts of function symmetry, even and odd function tests, and includes step-by-step algebraic verification. Ideal for students in Grades 9-11.

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