Let's analyze each function shown in the image to determine if it is even, odd, or neither:

1. Function $r$ (Top-left graph):

   • The function is symmetric with respect to the $y$-axis. This means that for any $x$, the function has $f(x) = f(-x)$.
   • This is the defining property of an even function.
   • Answer: Even

2. Function $s$ (Top-right graph):

   • The function is neither symmetric with respect to the $y$-axis nor the origin. It doesn't satisfy either $f(x) = f(-x)$ (even) or $f(x) = -f(-x)$ (odd).
   • Answer: Neither

3. Function $g(x) = -6x^4 + 5x^2$ (Bottom-left):

   • Both terms in the expression have even powers of $x$ (4 and 2), which means for any $x$, $g(-x) = g(x)$.
   • This satisfies the condition for an even function.
   • Answer: Even

4. Function $h(x) = 4x^5$ (Bottom-right):

   • The term has an odd power of $x$ (5), and $h(-x) = -h(x)$, which is the defining property of an odd function.
   • Answer: Odd

Summary:

• Function $r$: Even
• Function $s$: Neither
• Function $g(x) = -6x^4 + 5x^2$: Even
• Function $h(x) = 4x^5$: Odd

Let me know if you want more details or have any questions!

Here are five follow-up questions you might find interesting:

1. What is the general rule for identifying odd and even functions from their algebraic expressions?
2. Can a function be both even and odd? If so, what kind of function would that be?
3. How can the symmetry of a graph help in determining if a function is even or odd?
4. What would the graph of a function that is both even and odd look like?
5. Why do even functions have symmetry around the $y$-axis, and odd functions have rotational symmetry around the origin?

Tip: Always test a function by plugging in $f(-x)$ and comparing it to $f(x)$ to determine if it’s even, odd, or neither!