Determine whether the function is aven, odd, or neither Then determine whether the function's graph is symrnetric with respedt to the y-adis, the origin, or neither \( g(x)=x^{6}-4 x \) Determine whether the function is even, odd, or neither Choose the corred anewer below 0 odd 0 neither even

Let's analyze the function \( g(x) = x^{6} - 4x \) to determine its parity and the symmetry of its graph. ### 1. Determine if the Function is Even, Odd, or Neither **Definitions:** - **Even Function:** \( f(-x) = f(x) \) for all \( x \). - **Odd Function:** \( f(-x) = -f(x) \) for all \( x \). **Compute \( g(-x) \):** \[ g(-x) = (-x)^6 - 4(-x) = x^{6} + 4x \] **Compare \( g(-x) \) with \( g(x) \) and \( -g(x) \):** - **Even Check:** \[ g(-x) = x^{6} + 4x \neq g(x) = x^{6} - 4x \] So, \( g(x) \) is **not even**. - **Odd Check:** \[ -g(x) = -x^{6} + 4x \neq g(-x) = x^{6} + 4x \] So, \( g(x) \) is **not odd**. **Conclusion:** The function \( g(x) = x^{6} - 4x \) is **neither** even nor odd. ### 2. Determine the Symmetry of the Graph - **Symmetry with Respect to the y-axis:** This occurs if the function is even. Since \( g(x) \) is not even, there is **no y-axis symmetry**. - **Symmetry with Respect to the Origin:** This occurs if the function is odd. Since \( g(x) \) is not odd, there is **no origin symmetry**. **Conclusion:** The graph of \( g(x) \) has **no symmetry** with respect to the y-axis or the origin. ### Final Answer - **Parity:** Neither even nor odd. - **Graph Symmetry:** Neither symmetric with respect to the y-axis nor the origin. **Option Selected:** Neither