Sketch the graph of x operatorname(sgn) x

To sketch the graph of \( x \operatorname{sgn} x \), we need to understand the behavior of the function \( \operatorname{sgn} x \).<br /><br />The function \( \operatorname{sgn} x \) is defined as follows:<br />- \( \operator} x = 1 \) if \( x > 0 \)<br />- \( \operatorname{sgn} x = 0 \) if \( x = 0 \)<br />- \( \operatorname{sgn} x = -1 \) if \( x < 0 \)<br /><br />Now, let's consider the function \( x \operatorname{sgn} x \):<br />- For \( x > 0 \), \( \operatorname{sgn} x = 1 \), so \( x \operatorname{sgn} x = x \cdot 1 = x \).<br />- For \( x = 0 \), \( \operatorname{sgn} x = 0 \), so \( x \operatorname{sgn}cdot 0 = 0 \).<br />- For \( x < 0 \), \( \operatorname{sgn} x = -1 \), so \( x \operatorname{sgn} x = x \cdot (-1) = -x \).<br /><br />Therefore, the graph of \( x \operatorname{sgn} x \) consists of three line segments:<br />- A line segment with a positive slope for \( x > 0 \), where the function is equal to \( x \).<br />- A horizontal line at \( y = 0 \) for \( x = 0 \).<br />- A line segment with a negative slope for \( x < 0 \), where the function is equal to \( -x \).<br /><br />In summary, the graph of \( x \operatorname{sgn} x \wise linear function with a positive slope for \( x > 0 \), a horizontal line at \( y = 0 \) for \( x = 0 \), and a negative slope for \( x < 0 \).