Question

determine whether the function is even, odd, or neither. (recall the definitions of even and odd functions.)
f(x)=x\sin^{5}(x)

even

odd

neither

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Explanation:

Step1: Recall function - property definitions

An even function satisfies $f(-x)=f(x)$, and an odd function satisfies $f(-x)=-f(x)$. We need to find $f(-x)$ for $f(x)=x\sin^{5}(x)$.

Step2: Calculate $f(-x)$

Substitute $-x$ into the function: $f(-x)=(-x)\sin^{5}(-x)$. Since $\sin(-x)=-\sin(x)$, then $\sin^{5}(-x)=(-\sin(x))^{5}=-\sin^{5}(x)$. So $f(-x)=(-x)\times(-\sin^{5}(x))=x\sin^{5}(x)$.

Step3: Compare with definitions

Since $f(-x)=f(x)$, the function is even.

Answer:

even