Determine whether the function is even, odd, or neither. Then determine whether the functio \( g(x)=x^{2}+4 \mathrm{x} \) 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, \( \square \) neither \( \square \) y-axis origin

To determine whether \( g(x) = x^2 + 4x \) is even, odd, or neither, we need to check the properties of the function. First, compute \( g(-x) \): \[ g(-x) = (-x)^2 + 4(-x) = x^2 - 4x \] Since \( g(-x) \neq g(x) \) and \( g(-x) \neq -g(x) \), the function is neither even nor odd. In terms of symmetry, since the function does not satisfy \( g(-x) = g(x) \), it is not symmetric with respect to the \( y \)-axis. Likewise, it does not satisfy \( g(-x) = -g(x) \), so it is not symmetric with respect to the origin either. Thus, the function exhibits neither symmetry. This function's quadratic nature means it forms a parabola, but the presence of the \(4x\) term shifts the vertex away from the \(y\)-axis, confirming its asymmetric character. To visualize, you could plot \( g(x) \) and see the skew on one side! If you want to dive deeper into how symmetry in functions relates to their graphical appearances, check out resources on even and odd functions, such as "Calculus: Early Transcendentals" by James Stewart. Or for a playful exploration, you might enjoy "Math Doesn't Suck" by Danica McKellar, which offers delightful insights into various math concepts, including symmetry!