From the image you uploaded, we are tasked with determining whether the given functions are even, odd, or neither based on their graphs or equations.

1. The function $r$ (graph on the left):

• The graph of $r$ is a straight line with a negative slope, indicating a linear function.
• A function is odd if $f(-x) = -f(x)$, and linear functions of the form $f(x) = mx + b$ (where $m \neq 0$ and $b = 0$) are odd.
• Since the line passes through the origin and has a negative slope, this function is odd.

2. The function $s$ (graph on the right):

• The graph of $s$ appears symmetric with respect to the y-axis.
• A function is even if $f(-x) = f(x)$. This symmetry suggests that $s(x)$ satisfies the condition for being even.
• Therefore, the function $s$ is even.

3. The function $g(x) = -2x^5 + 3x^2$:

• To determine if the function is even, odd, or neither, check its terms:
  • $x^5$ is an odd power, and $x^2$ is an even power.
  • The term $-2x^5$ is odd, and $3x^2$ is even.
• Since this function contains both even and odd powers of $x$, it is neither even nor odd.

4. The function $h(x) = -6x^5 + 3x^3$:

• Both terms in $h(x)$ are odd powers of $x$ (i.e., $x^5$ and $x^3$).
• A function composed entirely of odd powers with coefficients behaves like an odd function, as $f(-x) = -f(x)$ holds for each term.
• Therefore, the function $h(x)$ is odd.

Summary:

• Function $r$: Odd
• Function $s$: Even
• Function $g(x) = -2x^5 + 3x^2$: Neither
• Function $h(x) = -6x^5 + 3x^3$: Odd

Would you like more details on any of these? Here are some related questions:

1. How can you determine if a function is even from its equation alone?
2. What are the geometric properties of even and odd functions?
3. Why is a function with mixed even and odd powers (like $g(x)$) considered neither even nor odd?
4. How do transformations like shifts affect whether a function is even or odd?
5. What is the significance of symmetry in determining whether a function is even or odd?

Tip: Always check the behavior of a function by substituting $x$ with $-x$ to determine if it's even, odd, or neither!