Question

1. 函数$y = xsin x$的部分图象是（）

a.
\

$$\begin{tikzpicture} \\draw-> (-3,0) -- (3,0) noderight {$x$}; \\draw-> (0,-1) -- (0,2) nodeabove {$y$}; \\drawthick, smooth, domain=-3:3 plot (x, {abs(x)*sin(x r)}); \\fill (0,0) circle (1pt); \\end{tikzpicture}$$

b.
\

$$\begin{tikzpicture} \\draw-> (-3,0) -- (3,0) noderight {$x$}; \\draw-> (0,-2) -- (0,2) nodeabove {$y$}; \\drawthick, smooth, domain=-3:3 plot (x, {x*sin(x r)}); \\fill (0,0) circle (1pt); \\end{tikzpicture}$$

c.
\

$$\begin{tikzpicture} \\draw-> (-3,0) -- (3,0) noderight {$x$}; \\draw-> (0,-2) -- (0,2) nodeabove {$y$}; \\drawthick, smooth, domain=-3:3 plot (x, {-abs(x)*sin(x r)}); \\fill (0,0) circle (1pt); \\end{tikzpicture}$$

d.
\

$$\begin{tikzpicture} \\draw-> (-3,0) -- (3,0) noderight {$x$}; \\draw-> (0,-2) -- (0,2) nodeabove {$y$}; \\drawthick, smooth, domain=-3:3 plot (x, {-x*sin(x r)}); \\fill (0,0) circle (1pt); \\end{tikzpicture}$$

Explanation:

Step1: 判断函数的奇偶性

对于函数\(y = f(x)=x\sin x\)，其定义域为\(R\)，关于原点对称。计算\(f(-x)\)：\(f(-x)=(-x)\sin(-x)\)，根据三角函数的性质\(\sin(-\alpha)=-\sin\alpha\)，则\(f(-x)=(-x)\times(-\sin x)=x\sin x = f(x)\)，所以函数\(y = x\sin x\)是偶函数，其图象关于\(y\)轴对称。选项B和D的图象不关于\(y\)轴对称，所以排除B、D。

Step2: 分析\(x\in(0,\pi)\)时函数的取值

当\(x\in(0,\pi)\)时，\(x>0\)，\(\sin x>0\)（因为正弦函数在\((0,\pi)\)上的值大于\(0\)），根据乘法法则，同号相乘为正，所以\(y = x\sin x>0\)。选项C中当\(x\in(0,\pi)\)时，函数值小于\(0\)，不符合，所以排除C。

Answer:

A. 函数\(y = x\sin x\)的图象