The image contains three functions, $f(x)$, $g(x)$, and $h(x)$, and asks to determine if each function is even, odd, or neither.

1. For $f(x) = \frac{1}{4x^3}$:

   • A function is odd if $f(-x) = -f(x)$.
   • Let's test $f(-x)$: $f(-x) = \frac{1}{4(-x)^3} = \frac{1}{-4x^3} = -\frac{1}{4x^3} = -f(x)$
   • Since $f(-x) = -f(x)$, the function $f(x)$ is odd.

2. For $g(x) = 3x |x|$:

   • A function is odd if $g(-x) = -g(x)$.
   • Let's test $g(-x)$: $g(-x) = 3(-x) |-x| = 3(-x)(x) = -3x |x| = -g(x)$
   • Since $g(-x) = -g(x)$, the function $g(x)$ is also odd.

3. For $h(x) = \sqrt{8x^2 + 1}$:

   • A function is even if $h(-x) = h(x)$.
   • Let's test $h(-x)$: $h(-x) = \sqrt{8(-x)^2 + 1} = \sqrt{8x^2 + 1} = h(x)$
   • Since $h(-x) = h(x)$, the function $h(x)$ is even.

Summary:

• $f(x)$ is odd.
• $g(x)$ is odd.
• $h(x)$ is even.

Would you like further details or have any questions?

Here are 5 related questions:

1. How can we determine if a function is neither even nor odd?
2. What happens when a function is both even and odd?
3. Can a polynomial function of degree 3 be even?
4. How does the presence of absolute values affect the symmetry of a function?
5. What other methods can be used to verify if a function is even or odd?

Tip: For any rational or radical function, always check the behavior of the function under input sign changes to determine its symmetry properties.