Determine whether the function is even, odd, or neither. Then describe the symmetry. f(x)=xsqrt (5-x^2) even odd neither Symmetry: x=y symmetry no symmetry x-axis symmetry y-axis symmetry origin symmetry

To determine whether the function \( f(x) = x\sqrt{5 - x^2} \) is even, odd, or neither, we need to check the following properties:<br /><br />1. A function \( f(x) \) is even if \( f(-x) = f(x) \) for all \( x \) in the domain of \( f \).<br />2. A function \( f(x) \) is odd if \( f(-x) = -f(x) \) for all \( x \) in the domain of \( f \).<br /><br />Let's check these properties for \( f(x) \):<br /><br />First, compute \( f(-x) \):<br />\[ f(-x) = (-x) \sqrt{5 - (-x)^2} = (-x) \sqrt{5 - x^2} \]<br /><br />Now, compare \( f(-x) \) with \( f(x) \):<br />\[ f(x) = x \sqrt{5 - x^2} \]<br />\[ f(-x) = (-x) \sqrt{5 - x^2} \]<br /><br />Clearly, \( f(-x) \neq f(x) \), so the function is not even.<br /><br />Next, check if \( f(-x) = -f(x) \):<br />\[ -f(x) = -x \sqrt{5 - x^2} \]<br />\[ f(-x) = (-x) \sqrt{5 - x^2} \]<br /><br />Since \( f(-x) = -f(x) \), the function is odd.<br /><br />Therefore, the function \( f(x) = x\sqrt{5 - x^2} \) is odd.<br /><br />**Symmetry:**<br />Since the function is odd, it has origin symmetry. that the graph of the function is symmetric with respect to the origin.<br /><br />So, the correct answer is:<br />- odd<br />- Symmetry: origin symmetry