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 \(\boxed{\text{neither even nor odd, and is not symmetric to the y-axis nor the origin}}\).