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!