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The given function is \( g(x) = x^4 - 3x \).

We need to determine if this function is even, odd, or neither, and also check for symmetry with respect to the y-axis, the origin, or neither.

### Step 1: Check if the function is even
A function is **even** if \( g(-x) = g(x) \) for all \( x \).

Let's compute \( g(-x) \):
\[
g(-x) = (-x)^4 - 3(-x) = x^4 + 3x
\]
Since \( g(-x) = x^4 + 3x \) and \( g(x) = x^4 - 3x \), we see that \( g(-x) \neq g(x) \). Therefore, the function is **not even**.

### Step 2: Check if the function is odd
A function is **odd** if \( g(-x) = -g(x) \) for all \( x \).

We already know that:
\[
g(-x) = x^4 + 3x
\]
And:
\[
-g(x) = -(x^4 - 3x) = -x^4 + 3x
\]
Since \( g(-x) \neq -g(x) \), the function is **not odd**.

### Step 3: Conclusion (Even, Odd, or Neither)
Since the function is neither even nor odd, the correct answer is **neither**.

### Step 4: Symmetry
Now, let's check for symmetry with respect to the y-axis, the origin, or neither.

- **Symmetry with respect to the y-axis**: A function is symmetric with respect to the y-axis if it is even. Since the function is not even, it is **not symmetric with respect to the y-axis**.
  
- **Symmetry with respect to the origin**: A function is symmetric with respect to the origin if it is odd. Since the function is not odd, it is **not symmetric with respect to the origin**.

- **Neither**: Since the function is neither even nor odd, and has no symmetry with respect to the y-axis or origin, the correct answer is **neither**.

### Final Answer:
- The function is **neither** even nor odd.
- The graph of the function has symmetry with respect to **neither** the y-axis nor the origin.

Do you want more details or have any further questions?

Here are 5 related questions you can explore:
1. What conditions must a function satisfy to be classified as even?
2. How can you algebraically determine if a function is odd?
3. What are examples of functions that are both even and odd?
4. How does symmetry affect the shape of a function's graph?
5. What is the relationship between function symmetry and its graph's reflection?

**Tip:** Always start by testing symmetry through substitution \( g(-x) \), then check conditions for evenness and oddness!

Determine whether the function is even, odd, or neither. Then determine whether the function's graph is symmetric with respect to the y-axis, the origin, or neither. The function is g(x) = x^4 - 3x.

Learn how to determine if the function g(x) = x^4 - 3x is even, odd, or neither, and check its symmetry with respect to the y-axis, origin, or neither. This problem covers key algebraic concepts such as function symmetry and analysis. Suitable for students in Grades 11-12.

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