Questions: 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. g(x)=x^6-2x Determine whether the function is even, odd, or neither. Choose the correct answer below. - even - odd - neither Determine whether the graph of the function is symmetric with respect to the y-axis, the origin, or neither. Select all that apply. - neither - y-axis - origin

Solution Steps

To determine if the function $g(x) = x^6 - 2x$ is even, odd, or neither, we need to evaluate $g(-x)$ and compare it to $g(x)$.

• A function is even if $g(-x) = g(x)$.
• A function is odd if $g(-x) = -g(x)$.

Calculate $g(-x)$:

$$g(-x) = (-x)^6 - 2(-x) = x^6 + 2x$$

Now, compare $g(-x)$ with $g(x)$:

• $g(x) = x^6 - 2x$
• $g(-x) = x^6 + 2x$

Since $g(-x) \neq g(x)$ and $g(-x) \neq -g(x)$, the function is neither even nor odd.

To determine the symmetry of the graph, we consider the following:

• A graph is symmetric with respect to the $y$-axis if the function is even.
• A graph is symmetric with respect to the origin if the function is odd.

Since the function is neither even nor odd, the graph is neither symmetric with respect to the $y$-axis nor the origin.

Final Answer