What is the correct representation of the function that is the negative reciprocal of x?

The function that is the negative reciprocal of ( x ) can be understood by examining what a reciprocal is. The reciprocal of a number ( x ) is ( \frac{1}{x} ). Therefore, the negative reciprocal means taking the reciprocal of ( x ) and then negating that value.

When you take the reciprocal of ( x ), you get ( \frac{1}{x} ). By applying the negative sign, we end up with

[ y = -\frac{1}{x} ]

This is exactly what is represented by the option provided. The negative in front indicates that the function decreases as ( x ) increases, taking on negative values when ( x ) is positive and vice versa, which is characteristic of the negative reciprocal function.

The other options do not represent the negative reciprocal of ( x ). The option that suggests ( y = x ) simply shows a direct proportionality, while ( y = \frac{1}{x} ) shows the positive reciprocal. Lastly, ( y = -x² ) depicts a quadratic function that decreases for positive ( x ) but does not represent the concept of reciprocal, making it irrelevant here. Thus, the