To determine whether each function is even, odd, or neither, let's analyze them one by one based on the provided graphs and equations.

1. Function $r$ (Top Left Graph)

A function is even if it satisfies $f(-x) = f(x)$ and odd if $f(-x) = -f(x)$.

• Observing the graph, the function $r$ is symmetric about the y-axis.
• This symmetry indicates that $f(-x) = f(x)$, which is the definition of an even function.

Answer: Even

2. Function $s$ (Top Right Graph)

• To determine symmetry, we check if the function has rotational symmetry about the origin (odd) or y-axis symmetry (even).
• Observing the graph, the function $s$ has no symmetry about the y-axis or the origin.
• Therefore, it is neither even nor odd.

Answer: Neither

3. Function $g(x) = -6x^4 + 7x^2$

• For an algebraic function:
  • A function is even if all the exponents of $x$ are even.
  • A function is odd if all the exponents of $x$ are odd.
• In $g(x) = -6x^4 + 7x^2$, all powers of $x$ (4 and 2) are even.
• This confirms that $g(x)$ is even.

Answer: Even

4. Function $h(x) = -4x^5 + 7x^2$

• In $h(x) = -4x^5 + 7x^2$, we see:
  • The term $-4x^5$ has an odd power (5).
  • The term $7x^2$ has an even power (2).
• Since $h(x)$ contains a mix of odd and even powers, it cannot satisfy the conditions for being purely even or odd.
• Therefore, $h(x)$ is neither even nor odd.

Answer: Neither

Final Answers:

1. Function $r$: Even
2. Function $s$: Neither
3. Function $g(x)$: Even
4. Function $h(x)$: Neither

Let me know if you'd like further clarification on any of these!

Related Questions:

1. What is the algebraic test to check if a function is odd or even?
2. How does symmetry about the origin relate to odd functions?
3. Can a function be both even and odd? Why or why not?
4. How do polynomial exponents determine whether a function is even or odd?
5. How can transformations (e.g., shifts or reflections) affect function symmetry?

Tip:

Even functions have symmetry about the y-axis, while odd functions have rotational symmetry about the origin. Graphing can provide a quick visual confirmation!