https://file.aiquickdraw.com/mathbot/file/1725831545269x2avubs9.jpg

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.

Determine whether each function is even, odd, or neither for the following functions: f(x) = 1/(4x^3), g(x) = 3x |x|, h(x) = √(8x^2 + 1).

Learn how to determine if the functions f(x) = 1/(4x^3), g(x) = 3x |x|, and h(x) = √(8x^2 + 1) are even, odd, or neither. This problem covers key algebraic concepts and properties of even and odd functions, suitable for students in Grades 10-12.

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