Let's analyze and classify each function as even, odd, or neither.

Function 1: $f(x) = -2x^2 + 16x - 25$

To check if a function is:

• Even: $f(x) = f(-x)$
• Odd: $f(x) = -f(-x)$

1. Calculate $f(-x)$:

$f(-x) = -2(-x)^2 + 16(-x) - 25 = -2x^2 - 16x - 25$

2. Compare $f(-x)$ to $f(x)$:

• $f(-x) = -2x^2 - 16x - 25$
• $f(x) = -2x^2 + 16x - 25$

Since $f(-x) \neq f(x)$ and $f(-x) \neq -f(x)$, this function is neither even nor odd.

Function 2 (Graph):

The graph shows a cubic-like function with a curve, suggesting that we might be dealing with an odd function. Let's confirm using symmetry:

• Odd functions exhibit rotational symmetry about the origin.

Based on the graph, it appears to have symmetry around the origin, which suggests the function is odd.

Function 3 (Graph):

The graph displays a parabola, which typically suggests an even function since:

• Even functions exhibit symmetry about the y-axis.

Looking at the graph, the function appears symmetric about the y-axis, indicating that this function is even.

Function 4: $f(x) = x^5 + 3x^3$

1. Calculate $f(-x)$:

$f(-x) = (-x)^5 + 3(-x)^3 = -x^5 - 3x^3$

2. Compare $f(-x)$ to $f(x)$:

• $f(-x) = -x^5 - 3x^3$
• $f(x) = x^5 + 3x^3$

Since $f(-x) = -f(x)$, this function is odd.

Final Classification:

1. $f(x) = -2x^2 + 16x - 25$ → Neither
2. Graph (Cubic-like) → Odd
3. Graph (Parabola) → Even
4. $f(x) = x^5 + 3x^3$ → Odd

Would you like further details or explanations on any of the functions?

Follow-up Questions:

1. How can you prove that a function is odd using algebraic methods?
2. Can a function be both even and odd? Why or why not?
3. What characteristics of the graph help determine if a function is even or odd?
4. Are all polynomial functions either even or odd?
5. How does the degree of a polynomial affect its classification as even or odd?

Tip:

Always check the symmetry of the function's graph to quickly identify whether it is even, odd, or neither.