g(x)=x^{4}+3x
Determine if function is even, odd, or neither, and determine if the function is symmetric to the y-axis, the origin, or neither.

Step 1 ：We are given the function \(g(x) = x^4 + 3x\).

Step 2 ：We need to determine if this function is even, odd, or neither. A function is even if \(f(x) = f(-x)\) for all x in the function's domain. A function is odd if \(-f(x) = f(-x)\) for all x in the function's domain.

Step 3 ：Let's first check if the function is even. We substitute \(-x\) into the function and simplify: \(g(-x) = (-x)^4 + 3(-x) = x^4 - 3x\).

Step 4 ：We can see that \(g(-x)\) is not equal to \(g(x)\), so the function is not even.

Step 5 ：Next, let's check if the function is odd. We substitute \(-x\) into the function and simplify: \(-g(x) = -(x^4 + 3x) = -x^4 - 3x\).

Step 6 ：We can see that \(-g(x)\) is not equal to \(g(-x)\), so the function is not odd.

Step 7 ：Since the function is neither even nor odd, it is also neither symmetric to the y-axis nor the origin.

Step 8 ：Final Answer: The function \(g(x) = x^4 + 3x\) is \(\boxed{\text{neither even nor odd, and is not symmetric to the y-axis nor the origin}}\).